to show that under $\mathfrak{m}_{\mathcal{F}}>\omega_{1}$, there are no
Suslin trees. Baumgartner proved something stronger by using the cardinal
$\mathfrak{m}$. He proved that under $\mathfrak{m}>\omega_{1}$, every
Aronszajn tree is special. Thus, a natural question is the following:

######  Problem 6.2.3.

Let $\mathcal{F}$ be a $2$-capturing construction scheme. Does the inequality
$\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ implies that every Aronszajn tree is
special?

In some sense, the non-existence of Suslin trees under
$\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ says that the property of being
$2$-capturing is not strong enough to imply the existence of a Suslin tree. On
the other hand $FCA$ does imply the existence of a Suslin tree. So how much
capturing is it really needed for this result?

######  Problem 6.2.4.

Let $3\leqslant n\in\omega$ and $\mathcal{F}$ be a $3$-capturing construction
scheme. Does the inequality $\mathfrak{m}^{n}_{\mathcal{F}}>\omega_{1}$
implies Suslin’s hypothesis?

The same question applies to entangled sets.

######  Problem 6.2.5.

Does the existence of $2$-entangled sets follow from either $CA_{n}$ or $CA?$
In other words, which is the weakest capturing axiom implying the existence of
$2$-entangled sets?

###  6.3 More problems regarding applications

In Theorem 3.3.9 we proved that, from $CA_{3}$, we can construct a coloring
$c:[\omega_{1}]^{2}\longrightarrow 2$ which is $2$-bounded, has no uncountable
injective sets, but is $ccc$ destructible with respect to this last property.

######  Problem 6.3.1.

Does the existence of a coloring such as the one in Theorem 3.3.9 follow from
$CA_{2}$? Given a construction scheme $\mathcal{F}$, does
$\mathfrak{m}_{\mathcal{F}}>\omega_{1}$ imply the non-existence of such a
coloring?

Regarding Tukey types, we propose the following question.

######  Problem 6.3.2.

How many distinct Tukey types on $\omega_{1}$ exist under $CA_{2}$?

Particularly, we can ask:

######  Problem 6.3.3.

Suppose that $\mathcal{F}$ and $\mathcal{F}^{\prime}$ are distinct
$2$-capturing construction schemes. Find sufficient conditions under which
$(\mathcal{B}_{\mathcal{F}},\leqslant)$ is incomparable with
$(\mathcal{B}_{\mathcal{F}^{\prime}},\leqslant)$.

Under Martin’s Axiom, a great variety of ultrafilters can be constructed. Is
the same true for $\mathfrak{m}_{\mathcal{F}}$?

######  Problem 6.3.4.

Let $2\leqslant n\in\omega$ and $\mathcal{F}$ be an $n$-capturing construction
scheme. Does $\mathfrak{m}^{n}_{\mathcal{F}}>\omega_{1}$ imply the existence
of union ultrafilters?

Let $S\subseteq\omega_{1}$ be stationary. Recall that a ladder system on $S$
is a sequence $\langle L_{\alpha}\rangle_{\alpha\in S}$ such that $L_{\alpha}$
is an unbounded subset of $\alpha$ of order type $\omega$ for each $\alpha\in
S.$

######  Definition 6.3.5 (Uniformization of ladders).

A ladder system $\langle L_{\alpha}\rangle_{\alpha\in S}$ is said to be
uniformizable if of each sequence $\langle f_{\alpha}\rangle_{\alpha\in S}$ of
functions $f_{\alpha}:L_{\alpha}\longrightarrow\omega$, there is
$F:\omega_{1}\longrightarrow\omega$ such that:

$f_{\alpha}=^{*}f$

for any $\alpha\in S$.

It is known that $\mathfrak{m}>\omega_{1}$ implies that every ladder system is
uniformizable. On the other hand, $2^{\omega}<2^{\omega_{1}}$ implies that no
ladder system is uniformizable (see [27]). Throughtout the course of the
investigation of construction schemes, we tried to construct ladder systems
with interesting properties. In particular, we tried to construct a ladder
system which is no uniformizable by using $FCA$. Nevertheless we didn’t
succeed in that task. So:

######  Problem 6.3.6.

Does the existence of a non-uniformizable ladder system follow from $FCA$?
What about $FCA(part)$?

######  Problem 6.3.7.

Let $2\leqslant n\in\omega$ and $\mathcal{F}$ be an $n$-capturing construction
scheme. Does $\mathfrak{m}^{n}_{\mathcal{F}}>\omega_{1}$ imply that every
ladder system is uniformizable?

To learn more about uniformizability of ladder systems see [8] and [23].

###  6.4 Problems about construction schemes

Recently, in [88], Assaf Rinot and Roy Shalev introduced the principle
$\clubsuit_{AD}$. This principle follows both from $\clubsuit$ and from the
existence of a Suslin Tree. Thus, it also follows from $FCA$. In this way, we
may ask:

######  Problem 6.4.1.

Is any instance of $\clubsuit_{AD}$ a consequence of $CA_{n}$ for some
$2\leqslant n\in\omega?$ What about $CA_{n}(part)?$

One of the most famous problems outside set theory, in which set theory played
an important role, was the solution of Naimark’s problem regarding
$C^{*}$-algebras. In [5], Charles Akemann and Nik Weaver constructed a
counterexample to Naimark’s problem using $\Diamond$-principle. In [18],
Daniel Calderón and Ilijas Farah isolated the guessing principle
$\Diamond^{Cohen}$, which follows from $\Diamond$. There, they constructed a
counterexample to Naimark’s problem using $\Diamond^{Cohen}$.

######  Problem 6.4.2.

Is $CA_{n}$, $CA$ or $FCA$ a consequence of $\Diamond^{Cohen}$? What about
$CA_{n}(part)$, $CA(part)$ and $FCA(part)$?

######  Problem 6.4.3.

Does the existence of a counterexample to Naimark’s problem follow from
$FCA(part)$?

In [78], Mirna Džamonja, Michael Hrušák and Justin Moore introduced
parametrized diamonds, which are weakenings of $\Diamond$ that may hold in
models where $CH$ fails. Both capturing construction schemes and parametrized
diamonds are useful tool for building interesting objects of size
$\omega_{1}$. We ask:

######  Problem 6.4.4.

Is there any relation between $FCA$ (respectively $CA_{n}$) and any
parametrized $\Diamond$-principle?

It would be specially interesting to know if there is any realtionship between
$FCA$ and $\Diamond(non(\mathcal{M})).$

We already know that the forcing $\mathbb{C}_{\kappa}$ forces $FCA(part)$
whenever $\kappa$ is an uncountable ordinal. But what happens when we only add
one Cohen real?

######  Problem 6.4.5.

Does the Cohen forcing $\mathbb{C}$ force $FCA$? What about $FCA(part)$?

By far, the most interesting problem from the point of view of the author is
the following:

######  Problem 6.4.6.

Extend the theory of capturing construction schemes to higher cardinals.

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# Is a model equivalent to its computer implementation?

Beatrix C. Hiesmayr  Universität Wien, Fakultät für Physik, Währingerstrasse
17, Vienna, Austria  Marc-Thorsten Hütt  School of Science, Constructor
University Bremen, Germany

###### Abstract

A recent trend in mathematical modeling is to publish the computer code
together with the research findings. Here we explore the formal question,
whether and in which sense a computer implementation is distinct from the
mathematical model. We argue that, despite the convenience of implemented
models, a set of implicit assumptions is perpetuated with the implementation
to the extent that even in widely used models the causal link between the
(formal) mathematical model and the set of results is no longer certain.
Moreover, code publication is often seen as an important contributor to
reproducible research, we suggest that in some cases the opposite may be true.
A new perspective on this topic stems from the accelerating trend that in some
branches of research only implemented models are used, e.g., in artificial
intelligence (AI). With the advent of quantum computers we argue that
completely novel challenges arise in the distinction between models and
implementations.

##  I Introduction

The use of mathematical models 111In this paper we do not question the
different views on what mathematics is such as a Platonists’, formalists’,
logicists’, intuitionists, empiricists’ or structuralists’ views, rather we
purely are interested in how it is used in today’s science. has become
increasingly commonplace in many fields, including biology, physics,
economics, sociology and engineering. Mathematical models are abstract
representations of systems or processes that allow researchers to study their
behavior under different constraints, understand the mechanisms behind a set
of observations, and, finally, to make predictions. Those predictions are then
typically compared with observation in the real world. Often, the basic models
are based on observations from the past.

Is a model equivalent to its computer implementation? The short answer is, of
course, no. It starts by the well-known fact that any number, such as for
instant $1$, can only be represented by a finite amount resources on a
computer (float, double). Those limitations are well known and studied and
have the awareness of modelers and users. However, a detailed reflection of
this matter is mandatory, because:

  * (1)

Reflecting on this distinction between a model and its implementation offers
insight into our view on mathematical models, how we represent them and how
this changes over time.

  * (2)

Studying the equivalences (or lack thereof) of mathematical models and their
implementations can allow us to understand some core reasons for the lack of
reproducibility of computational results (see, e.g., [2, 3, 4]).

  * (3)

The topic draws attention to recent trends and innovations in academic
publishing, which shape future research.

Code submission for mathematical or computational models has become a frequent
requirement of funding agencies and academic journals alike. However, in the
literature discussing this requirement [5, 6, 7, 8] the differences between
model and code highlighted here, as well as the potential disadvantages of
resorting solely to implemented models, are hardly discussed.

A model is formulated on the level of equations (or any mathematical language
appropriate for the system at hand). It is based on explicit and implicit
assumptions and mostly allows for some consistence checks or proofs, for
instance a result can be only a positive number and never a negative or
complex number. Mathematical models are abstract formulations derived from
human thought processes. They come in the form of relations between numbers,
in the form of formulas or a set of equations. Often, those equations cannot
be studied fully analytically (i.e., just in its original functional form). As
a consequence, the original description is mathematical, but implementations
of such a model are created for the purpose of running computer simulations or
evaluating the model beyond what is analytically possible.

An implicit assumption of a ’good’ computer implementation is that the results
are a consequence of the mathematical model and independent of the model’s
specific implementation. This also means that any aspect pertaining solely to
the computer implementation of the model is irrelevant for the model behavior
and for the results obtained with the model. Examples of such implementation
details are the choice of initial conditions, the sequence, in which
operations formalized in the model are executed, and the numerical recipes
employed for solving or simulating the model.

In some fields of science, the strength of mathematical modeling is to provide
precise quantitative predictions of some measurable quantities (see also Fig.
2). In these cases, the model stems from a theory that claims to represent
this aspect of the world with maximal fidelity. The quality of an
implementation based on the mathematical model, which in itself is based on
some theoretical model, is then typically quantified by the closeness of the
predicted quantity to the measurement quantity. Even if successful in this
sense, the application to another case may lead to discrepancies, requiring a
change in the implementation or its underlying model.

In other cases a qualitative agreement with observations is targeted, as in
the case of modeling socioeconomic systems [9, 10] or other complex systems
[11, 12, 13]. Then the claim of the modeling effort is that the model contains
all the mechanistic ingredients to capture a certain (often counter-intuitive)
phenomenon. In these fields of science particular importance is given to the
smallest, most minimal models capturing an obverved behavior (smallest in the
sense of containing the smallest number of degrees of freedom and/or the
smallest number of parameters). Such ’toy models’ or minimal models are a
cornerstone of many applications of, e.g., statistical physics [14, 15, 16].

With this perspective paper we want to draw attention to the non-trivial and
crucial relationship between a mathematical model and its computer
implementation and the subtle and informative ways, in which they can be
different. But we also want to emphasize some potential challenges, which come
along with the otherwise commendable trend in academic publishing of requiring
code submissions for mathematical models [4, 17], because we feel that the
current debate at times underestimates the slight distortions of the modeling
landscape that come along with it.

##  II  A detailed assessment

In this section we illustrate our general point about the impact of code
availability for mathematical models on model diversity and provide more
details about the distinction of models from their implementation.

###  II.1 Technical issues in the implementation

Re-implementations decouple the information flow from the flow driven by small
errors and implementation design decisions not covered by the original
mathematical model. As mentioned before, among these are discretization
effects, choices of initializations, tie-breaking criteria, the order of
executing logical steps and many more. Figure 1 is a schematic illustration of
this change in information flow (with information flow given in green and flow
of errors and implementation design decisions given in red).

Figure 1:  Schematic representation of the difference between (a) model
implementations published along with the academic results and (b) the
publication serving as the main source of information about the mathematical
model. In (a) information flow and the flow of errors or the impact of design
decisions, which are not part of the original mathematical model, are coupled,
while in (b) due to the diversity of implementations by different modelers
those decisions by a single modeler are not propagated.

As an early example of the challenging relationship between a model and its
computer implementation, Edward Lorenz’s numerical experiments with a
$12$-variable version of his weather model in the early 1960s come to mind.
The rounding errors of the computer (providing initial conditions only to the
third decimal in the output) were enough to lead to drastically different
simulation runs [18]. These observations – and Lorenz’s ingenuity of
interpreting them – culminated in the discovery of a deterministic model
represented as coupled ordinary differential equations (ODEs) – the Lorenz
equations – in his seminal paper in 1963 [19].

Some practical aspects also affect this debate. Model implementations can be
platform dependent (using packages, which change over time) or dependent on
the numerical recipes (e.g., ODE solvers) used. In the computer algebra and
programming environment Mathematica, for example, a matrix consisting of
entries $1/2$ and a matrix consisting of entries $0.5$ can yield markedly
different results in terms of representation, normalization and numerical
accuracy, as the first variant triggers the symbolic routines, while the
second triggers the numerical routines of this programming environment.

Summarizing, the list, in which a model can differ from its technical
implementation, is quite long. It includes factors like initialization,
discretization (e.g., within a solver), implementation errors, statistics
(length of the simulation, transient, number of simulations), tiebreaking
criteria, order of update rules, noise implementation (and quality of the
random number generator) and many more.

###  II.2 Reproducibility – a key argument for the quantity of research?

What can be learned from discussing the relationship between a model and its
implementation? The best discussed aspect of this topic is certainly the
reproducibility of scientific results. In Ref. [4], for example, published
(and electronically available) models have been assessed regarding their
capability to produce the results of the core publication behind each model.
The striking observation in this field of biological modeling, namely how
limited this direct reproducibility is, has led to a range of recommendations
regarding the publication of model implementations and the curation of
mathematical models. A strong argument in favor of this general procedure is
provided by evidence that reproducible models receive more citations [20].

Other aspects of these topics are general error propagation and the
propagation of implicit knowledge beyond the original model equations. As a
trivial example, to select one element among the techical issues discussed
above, a particular way of choosing initial conditions for a system of coupled
differential equations hard-wired into the implementation might prevent a full
numerical exploration of the dynamical scope of the model, unless the
implementation is altered in this respect or the model is re-implemented. Re-
implementing existing mathematical model is also of high educational value and
is often a good starting point for a young graduate student. Moreover, it is a
kind of quality check between implementation and mathematical model (by which
errors have often been identified in the past, even by models widely used.

Note that here our focus is on highlighting a potential danger in this
practice of required / recommended code publication. It needs to be
emphasized, however, that indeed availability of implemented models has hugely
amplified model (re-)usage and comparison of data with models.

##  III  Beyond standard mathematical models: AI and quantum computers

###  III.1 Data-driven modeling: AI

Some insight in the relationship between a model and its implementation can be
gained by imagining to discard the mathematical model completely and rather
directly implement all systemic knowledge in the form of a computer program.
It then becomes obvious that, indeed, the formulation of a mathematical model
and the design of a computer implementation are also conceptually quite
different tasks. If you strive for an implementation only, the choice of
functions (e.g., linear, quadratic, sigmoidal, …) and conciseness may be less
relevant. If you strive for a mathematical model, elegance and simplicity are
important factors [21]. Conceptual mistakes are more easily spotted in
equations than in their implementations.

Artificial intelligence (AI) is an example of such a purely data-driven
approach to modeling. There, the implementation task is fully given to the
training process of the device using available data, and the distinction
between model and implementation becomes impossible. We all experience
currently the experiment how far this approach can be taken and which social-
economic developments will be triggered.

Both avenues of modeling, theory-driven and data-driven, have given rise to
modern forms of computation, quantum computers in the theory-driven case, AI
in the data-driven case. Figure 2 summarizes this situation.

Figure 2:  Summary of the concepts driving the distinction between models and
their implementation. Left: Standard (historical) approach, where theory-
driven mathematical models are explored on their own and with the help of
computer simulations. This implements a cycle model $\rightarrow$
implementation $\rightarrow$ prediction $\rightarrow$ observations
$\rightarrow$ model, which also challenges, and eventually contributes to,
theory. The largest impact of this approach on modern computation is in the
form of quantum computers. Right: New trend, where observations are directly
translated into implementations (sometimes via a mediating mathematical model,
which however stays in the background, because the implementation is regarded
as the main outcome of the scientific endeavor). This process creates
predictions, which can be compared to new observations. The largest impact of
this approach on modern computation is in the form of artificial intelligence
(AI).

After briefly highlighting the role of AI in this discussion of the
distinction between a model and its implementation, we will in the following
section discuss the other cornerstone of modern computation, quantum
computers.

###  III.2 Quantum computers

Quantum computers challenge the classical distinction between model and
implementation in a multitude of ways. In the following we will briefly
address this novel form of computation and discuss it in the context of models
and their implementation.

With the advent of quantum computers novel computing devices will be on the
market that will definitely “compute” or transfer information in a very
different way than classical computers. Here the implementation is even be
much more different since the quantum mechanical theory behind a given
algorithm is much deeper than in the classical case. Let us explain this in
more details.

Let us consider the easiest case, the encoding of a bit, i.e. “0” or “1”.
Classically, we need a system that can be in two alternative states, for which
we define one as “0” and the other as “1”. The reading device needs to be able
to distinguish the two cases (in the optimal case without any error). In a
quantum computer the bit information needs to be encoded in its building
blocks, e.g. in a quantum mechanical systems that “answers” by a given
measurement device –in the easiest case– with a dichotomic answer, which again
will be defined as “0” and “1”.

In strong contrast to classical systems, the states allowed by any such
quantum system is given by 222Without loss of generality we consider only pure
states, i.e. states to which no classical uncertainty is added.

$|\psi\rangle=\cos\frac{\theta}{2}\;|\textrm{``0''}\rangle+\sin\frac{\theta}{2}\cdot
e^{i\phi}\;|\textrm{``1''}\rangle$ (1)

which is called a “quantum bit” or shortly “qubit”. It contains in a sense
infinite information since the values $\theta,\phi$ can be taken from an
infinite set of numbers ($\theta\in[0,\pi]$ and $\phi\in[0,2\pi\\}$).
Consequently, a quantum state can be initiated in infinite many different
states (by choosing different $\theta,\phi$), whereas a classical computer has
either voltage $0$ (state “0”) or voltage $5$ (state “1”). However, one needs
to measure a quantum state, to obtain information about the initialization.
And herewith, the information is a one bit information, either “0” or “1”. Of
course, this is the same for classical computer, you need to use e.g. a
Voltmeter to measure the voltage.

So, what role do $\theta,\phi$ play? Those reveal, when one repeats the
procedure of initialisation and measuring. Then from this (infinite)
collection of measurements one will find that in a series of “0” occur with
probability $\cos^{2}\frac{\theta}{2}$ and “1” with the probability
$\sin^{2}\frac{\theta}{2}=1-\cos^{2}\frac{\theta}{2}.$ (The parameter $\phi$
only reveals if the state is rotated by some device or a different measurement
than “0” and “1” is chosen).

So the point is that on the theory level, we have the full information
($\theta,\phi$) since we can and must choose them for the initiation, but,
depending on the chosen experimental setup, we get only partial information
about the values. In our chosen setup only information about $\theta$ and not
$\phi$. Moreover, only if we have a repetition of the process, initialization
and measurement, we get successively access to this information, the quantum
theory is a probabilistic one.

To sum up, any theoretical model of a quantum mechanical process contains in
general more parameters than can be read out in an experiment and at least
some repetition, during which one has to imply the hypothesis that no
processes take place which would change the observable of interest, is needed
to access experimental results. Nowadays, the main issue in the simulation of
a quantum process on a classical computer is that a classical computer cannot
easily process all parameters, thus short cuts have to be implemented in
addition to all the problems listed in the previous sections.

A striking other problem in simulation quantum processes comes with
entanglement, which has no classical counterpart. It needs in a classical
simulation that parts of computations do depend on each other in a not
neccessarily causal way.

Running a quantum algorithm on a quantum computer, on the other hand, solves
this problem, however, our current quantum devices struggle with experimental
errors (decoherence), for which we so far have not found any simple way to
correct for.

##  IV Conclusion

In most cases, computer implementations of mathematical models are necessary
to actually use those models for prediction or simulation. Therefore, while
mathematical models and their computer implementations are technically
distinct objects, they are functionally intertwined in many applications. The
use of software to solve mathematical models allows for a more practical
usage, enabling higher model complexity real-world predictions and
applications. Despite their functional interdependence, it is important to
understand the distinctions between mathematical models and their computer
implementations. Mathematical models are conceptual and theoretical frameworks
that describe the behavior of systems, while computer implementations are
tangible tools used to simulate or predict system behavior based on those
theoretical frameworks.

As so eloquently summarized in [23]:

> Until only a decade or two ago, anyone pursuing this kind of activity
> [numerical simulations of mathematical models] had to have a solid grounding
> in mathematics. And that meant that such studies were done by people who had
> some idea, at an intuitive level, of how the original assumptions related to
> the emerging graphical display or other conclusions on their computer.

We believe that this process is drastically accelerated (and its implicit
dangers are substantially amplified) by the ’implementation monoculture’
enabled by the distribution of readily implemented mathematical models.

Implementations of the same mathematical model are not identical. And from our
perspective, understanding is mediated by analyzing equations (often with the
help of numerical simulations), not just by running the computer code.

We acknowledge the usefulness of model databases (like BioModels, [24]), in
particular in allowing experimentalists the means of directly comparing their
experimental findings with available mathematical models. We feel, though,
that there is a serious danger in the resulting ’implementation monoculture’
and we suggest to incentivize the reimplementation of existing mathematical
models, in order to help distinguish between universal results of a
mathematical model and those requiring additional implicit ingredients
included in a given implementation.

###### Acknowledgements.

The starting point of this paper is a discussion in July 2022 in Salzburg at
the Festive Session 2022 of the European Academy of Sciences and Arts. B.C.H.
acknowledges gratefully that this research was funded in whole, or in part, by
the Austrian Science Fund (FWF) project P36102-N. M.T.H. thanks the Hamburg
Institute for Advanced Study (HIAS) for hospitality and the Joachim Herz
Stiftung for funding his fellowship at HIAS, during which part of this work
has been performed.

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# Safe and Efficient Model Predictive Control Using Neural Networks: An
Interior Point Approach

Daniel Tabas and Baosen Zhang This work is partially supported by the National
Science Foundation Graduate Research Fellowship Program under Grant No.
DGE-1762114, NSF grants ECCS-1930605 and ECCS-2023531. Any opinions, findings,
conclusions, or recommendations expressed in this material are those of the
authors and do not necessarily reflect the views of the National Science
Foundation.Authors are with the department of Electrical and Computer
Engineering, University of Washington, Seattle, WA, United States.
$\\{$dtabas<EMAIL_ADDRESS>
###### Abstract

Model predictive control (MPC) provides a useful means for controlling systems
with constraints, but suffers from the computational burden of repeatedly
solving an optimization problem in real time. Offline (explicit) solutions for
MPC attempt to alleviate real time computational challenges using either
multiparametric programming or machine learning. The multiparametric
approaches are typically applied to linear or quadratic MPC problems, while
learning-based approaches can be more flexible and are less memory-intensive.
Existing learning-based approaches offer significant speedups, but the
challenge becomes ensuring constraint satisfaction while maintaining good
performance. In this paper, we provide a neural network parameterization of
MPC policies that explicitly encodes the constraints of the problem. By
exploring the interior of the MPC feasible set in an unsupervised learning
paradigm, the neural network finds better policies faster than projection-
based methods and exhibits substantially shorter solve times. We use the
proposed policy to solve a robust MPC problem, and demonstrate the performance
and computational gains on a standard test system.

##  I Introduction

Model predictive control (MPC) [1] is a powerful technique for controlling
systems that are subject to state and input constraints, such as agricultural
[2], automotive [3], and energy systems [4]. However, many applications
require fast decision-making which may preclude the possibility of repeatedly
solving an optimization problem online [5].

A popular approach for accelerating MPC is to move as much computation offline
as possible [5, 6]. These techniques, known as explicit MPC, involve
precomputing the solution to the MPC problem over a range of parameters or
initial conditions. Most of the research effort has focused on problems with
linear dynamics and constraints, and linear or quadratic cost functions. In
this case, the explicit MPC solution is a piecewise affine (PWA) function
defined over a polyhedral partition of the state constraints. However, many of
the applications of interest have cost functions that are not necessarily
linear or quadratic, or even convex. Further, the memory required to store the
partition and affine functions can be prohibitive even for modestly-sized
problems.

In order to reduce the complexity of explicit MPC, the optimal offline
solution can be approximated. Approximations generally fall into two
categories: partition-based solutions [7, 8, 9] that generate piecewise
control laws over coarser state space partitions, and learning-based solutions
[10, 11, 12, 13] that use function approximation to compactly represent the
optimal MPC policy. In this paper, we focus on the latter with the key
contribution of ensuring constraint satisfaction while exploring all feasible
policies.

Constraint satisfaction is crucial in many engineering applications, and the
ability of MPC to enforce constraints is a major factor in its popularity.
However, it is not straightforward to guarantee that a learning-based solution
will satisfy constraints. The main challenge arises from the fact that while
neural networks can limit their outputs to be in simple regions, there is no
obvious way of forcing complex constraint satisfaction at the output. In [10,
14], supervised and unsupervised learning were used to approximate the
solution of MPCs, but did not provide any feasibility guarantees. By contrast,
[11] trains an NN using a policy gradient approach, and guarantees feasibility
by projecting the NN output into the feasible action set. However, this extra
optimization step slows down the speed of online implementation, making it
difficult to use in applications that require high-frequency solutions [15].
Supervised learning approaches that provide safety guarantees [13, 12] rely on
a choice of MPC oracle that is not obvious when persistent disturbances are
present.

In this paper, we propose an NN architecture for approximating explicit
solutions to finite-horizon MPC problems with linear dynamics, linear
constraints, and arbitrary differentiable cost functions. The proposed
architecture guarantees constraint satisfaction without relying on projections
or MPC oracles. By exploring the _interior_ of the feasible set, we
demonstrate faster training and evaluation, and comparable closed-loop
performance relative to other NN architectures.

The proposed approach has parallels in interior point methods for convex
optimization [16]. Interior point methods first solve a _Phase I_ problem to
find a strictly feasible starting point. This solution is used to initialize
the _Phase II_ algorithm for optimizing the original problem. Our approach
accelerates both phases. The Phase I solution is given by a simple function
(e.g., affine map) and the Phase II problem is solved using an NN architecture
that can encode arbitrary polytopic constraints (Fig. 1).

The Phase II solution builds on a technique first proposed in [17], which uses
a _gauge map_ to establish equivalence between compact, convex sets. With
respect to [17], the current work has three novel aspects. First, the
reinforcement learning (RL) algorithm in [17] only uses information about the
constraints, and does not use information about the cost function or dynamics.
The resulting policy is safe, but can exhibit suboptimal performance. The MPC
formulation in the current paper gives rise to a training algorithm that can
exploit knowledge about the system, improving performance. Second, the MPC
formulation permits explicit consideration for future time steps. The RL
formulation cannot optimize entire trajectories due to the presence of
constraints. This inability to “look ahead” again limits the performance of
the RL algorithm. Finally, the previous work required a Phase I that used a
linear feedback to find a strictly feasible point. A linear feedback, however,
may not exist for some problems. The current work proposes a more general
class of Phase I solutions (piecewise affine), while providing a way to manage
the complexity of the Phase I solution.

Figure 1: Illustration of the interior point approach to learning-based MPC.
The set $\mathcal{F}(x_{0})$ represents the MPC feasible set, while
$\mu_{0}(x_{0})$ and $\mu_{\theta}(x_{0})$ are control input sequences
representing solutions to the Phase I and Phase II problems, respectively. The
neural network $\mu_{\theta}$ moves the Phase I solution to a more optimal
solution.

We demonstrate the effectiveness of the proposed technique on a 3-state test
system, and compare to standard projection- and penalty-based approaches for
learning with constraints. The results show that the proposed technique
achieves Pareto efficiency in terms of closed-loop performance and online
computation effort. All code is available at github.com/dtabas/gauge_networks.

Notation: The $p$-norm ball for $p\geq 1$ is
$\mathbb{B}_{p}=\\{z\mid\|z\|_{p}\leq 1\\}$. A polytope
$\mathcal{P}\subset\mathbb{R}^{n}:=\\{z\in\mathbb{R}^{n}\mid Fz\leq g\\}$ is
the (bounded) intersection of a finite number of halfspaces. Scaling of
polytopes by a factor $\lambda>0$ is defined as $\lambda\mathcal{P}=\\{\lambda
z\in\mathbb{R}^{n}\mid Fz\leq g\\}=\\{z\in\mathbb{R}^{n}\mid Fz\leq\lambda
g\\}$. Given a matrix $F$ and a vector $g$, the $i$th row of $F$ is denoted
$F^{(i)T}$ and the $i$th component of $g$ is $g^{(i)}$. The interior of any
set $\mathcal{Q}$ is denoted $\textbf{int }\mathcal{Q}$. The value of a
variable $y$ at a time interval $t$ is denoted $y_{t}$. A state or control
trajectory of length $\tau$ is written as the vector
$\textbf{x}=\begin{bmatrix}x_{1}^{T},\ldots,x_{\tau}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{n\tau}$
or
$\textbf{u}=\begin{bmatrix}u_{0}^{T},\ldots,u_{\tau-1}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{m\tau}$.
The column vector of all ones is 1. The symbol $\circ$ denotes function
composition.

##  II Problem Formulation

In this paper, we consider the problem of regulating discrete-time dynamical
systems of the form

$\displaystyle x_{t+1}=Ax_{t}+Bu_{t}+d_{t}$ (1)

where $x_{t}\in\mathbb{R}^{n}$ is the system state at time $t$,
$u_{t}\in\mathbb{R}^{m}$ is the control input, and $d_{t}\in\mathbb{R}^{n}$ is
an uncertain input that captures exogenous disturbances and/or linearization
error (if the true system dynamics are nonlinear) [18]. We assume the pair
$(A,B)$ is stabilizable. The input constraints (actuation limits) are
$\mathcal{U}=\\{u\in\mathbb{R}^{m}\mid F_{u}u\leq g_{u}\\}$ while the state
constraints arising from safety-critical engineering considerations are
$\mathcal{X}=\\{x\in\mathbb{R}^{n}\mid F_{x}x\leq g_{x}\\}$.

We consider the problem of operating the system (1) using finite-horizon model
predictive control. The goal is to choose, given initial condition
$x_{0}\in\mathcal{X}$, a sequence of inputs u of length $\tau$ that minimizes
the cost of operating the system while respecting the operational constraints.

However, since the disturbances $d_{t}$ are unknown ahead of time, the
designer must carefully consider how to achieve both optimality and constraint
satisfaction. Robust MPC literature contains many ways to handle the presence
of disturbances in both the cost and constraints [19]. For example, the
certainty-equivalent approach [5] considers only the nominal system
trajectory, while the min-max approach [9] considers the worst-case
disturbance. Interpolating between these two extremes, the tube-based approach
[20] considers the cost of a nominal trajectory while guaranteeing that the
true trajectory satisfies constraints. A stochastic point of view in [21]
considers the disturbance as a random variable and minimizes the expected cost
while providing probabilistic guarantees for constraint satisfaction.

In most robust MPC formulations, the set of possible disturbances is modeled
as either a finite set, a bounded set, or a probability distribution [22]. In
this paper, we assume the disturbances lie in a closed and bounded set
$\mathcal{D}:=\\{d\in\mathbb{R}^{n}\mid F_{d}d\leq g_{d}\\}$. In order to
ensure constraint satisfaction, we operate the system within a robust control
invariant set (RCI) $\mathcal{S}\subseteq\mathcal{X}$, defined as a set of
initial conditions for which there exists a feedback policy in $\mathcal{U}$
keeping all system trajectories in $\mathcal{S}$, under any disturbance
sequence in $\mathcal{D}$ [23]. In our simulations, we used approximately-
maximal RCIs computed with the semidefinite program from [24].

With $\mathcal{S}:=\\{x\in\mathbb{R}^{n}\mid F_{s}x\leq g_{s}\\}$, we define
the target set $\mathcal{T}$ as $\\{x\in\mathbb{R}^{n}\mid x+d\in\mathcal{S},\
\forall\ d\in\mathcal{D}\\}=\\{x\in\mathbb{R}^{n}\mid
F_{s}x\leq\tilde{g}_{s}\\}$ where for each row $i$,
$\tilde{g}_{s}^{(i)}=g_{s}^{(i)}-\max_{d\in\mathcal{D}}F_{s}^{(i)T}d$ [23].
Any policy that maps $\mathcal{S}$ to $\mathcal{T}$ under the nominal dynamics
will map $\mathcal{S}$ to itself under the true dynamics, rendering
$\mathcal{S}$ robustly invariant. By constraining the nominal state to the
target set, robust constraint satisfaction is guaranteed for the first time
step. Since $\mathcal{S}$ is RCI, this is sufficient for keeping closed-loop
trajectories inside $\mathcal{S}$. Under this formulation, the MPC problem is
posed as follows, given initial state $x_{0}$:

$\displaystyle\min_{\textbf{u}}\sum_{k=0}^{\tau-1}l(x_{k},u_{k})+l_{F}(x_{\tau})$
(2a) $\displaystyle\text{subject to $\forall\ k$: }x_{k+1}=Ax_{k}+Bu_{k}$ (2b)
$\displaystyle x_{k+1}\in\mathcal{T}$ (2c) $\displaystyle u_{k}\in\mathcal{U}$
(2d)

where $l$ and $l_{F}$ are stage and terminal costs that are differentiable but
possibly nonlinear or even non-convex. Although (2) differs from the standard
tube-based approach, the techniques introduced in this paper can be applied to
a variety of MPC formulations.

In this paper, we seek to derive a safe feedback policy
$\pi_{\theta}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ that approximates the
explicit solution to (2) by first approximating the optimal control sequence
with a function $\mu_{\theta}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m\tau}$ and
then implementing the first action of the sequence in the closed loop. In
practice, any MPC policy implemented in closed loop must be stabilizing and
recursively feasible. Recursive feasibility is the property that closed-loop
trajectories generated by the MPC controller will not lead to states in which
the MPC problem is infeasible. This property is guaranteed when $\mathcal{S}$
is RCI [23]. If recursive feasibility is not guaranteed, then a backup
controller must be developed or a control sequence that is feasible for the
most immediate time steps can be used. There is suggestion in the literature
that the latter approach performs quite well in practice [25], but the
theoretical aspects remain open. In terms of stability, recursive feasibility
guarantees that trajectories will remain within a bounded set. Since this work
focuses on constraint satisfaction, we do not consider stricter notions of
stability.

##  III Phase I: Finding a Feasible Point

The feasible set of (2) is a polytope
$\mathcal{F}(x_{0})\subseteq\mathbb{R}^{m\tau}$, defined by the following
inequalities in u:

$\displaystyle H_{s}(M_{0}x_{0}+M_{u}\textbf{u})$
$\displaystyle\leq\tilde{h}_{s},$ (3a) $\displaystyle H_{u}\textbf{u}$
$\displaystyle\leq h_{u}$ (3b)

where $H_{s},H_{u},M_{0},M_{u},\tilde{h}_{s},$ and $h_{u}$ are block matrices
and vectors derived from the system dynamics and constraints. In this paper,
we assume that $\mathcal{F}(x_{0})$ has nonempty interior for all
$x_{0}\in\mathcal{S}$. Since the state constraints $\mathcal{S}$ form an RCI,
$\mathcal{F}(x_{0})$ is already guaranteed to be nonempty, and the assumption
of nonempty interior is only marginally more restrictive.

The gauge map technique introduced in [17] provides a way to constrain the
outputs of a neural network
$\mu_{\theta}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m\tau}$ to
$\mathcal{F}(x_{0})$ without a projection or penalty function, but
$\mathcal{F}(x_{0})$ must contain the origin in its interior. If this is not
the case, then we must temporarily “shift” $\mathcal{F}(x_{0})$ by subtracting
any one of its interior points. In this section, we discuss several ways to
reduce the complexity of finding an interior point.

We begin by considering the feasibility problem for the one-step safe action
set defined as $\mathcal{V}(x_{0})=\\{u\in\mathbb{R}^{m}\mid
u\in\mathcal{U},Ax_{0}+Bu\in\mathcal{T}\\},$ which is guaranteed to have an
interior point by the assumption on $\mathcal{F}(x_{0}).$ One way to find an
interior point of $\mathcal{V}(x_{0})$ is to minimize the maximum constraint
violation:

$\displaystyle\min_{u,s}s$ (4a) $\displaystyle\text{subject to:
}F_{s}(Ax_{0}+Bu)\leq\tilde{g}_{s}+s\textbf{1}$ (4b) $\displaystyle F_{u}u\leq
g_{u}+s\textbf{1}$ (4c)

which has an optimal cost $s^{*}\leq 0$ if $\mathcal{V}(x_{0})$ is nonempty,
and $s^{*}<0$ if $\mathcal{V}(x_{0})$ has nonempty interior [16]. To avoid
solving a linear program online during closed-loop implementation, the
solution to (4) can be stored as a piecewise affine (PWA) function
$\pi_{0}(x_{0}):\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ [7]. Although
solutions to multiparametric LPs can be demanding on computer memory, we take
advantage of the fact that feasibility problems have low accuracy
requirements: any suboptimal solution to (4) that achieves a cost $s<0$ for
all $x_{0}\in\mathcal{S}$ is acceptable.

###### Definition 1

A function $\pi_{0}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is said to solve
(4) if, for all $x_{0}\in\mathcal{S}$, the optimal cost of (4) is negative
when the decision variable $u$ is fixed at $\pi_{0}(x_{0})$.

Existing techniques for approximate multiparametric linear programming [26],
especially those that generate continuous solutions [27], can be used to
reduce the memory requirements of offline solutions to (4).

To show just how far one can go with reducing complexity, we will construct an
affine (rather than PWA) function that solves (4), for the system studied in
Section V. Let $\pi_{0}(x_{0})=Wx_{0}+w$. If $W\in\mathbb{R}^{m\times n}$ and
$w\in\mathbb{R}^{m}$ satisfy

$\displaystyle F_{x}(Ax_{0}+B(Wx_{0}+w))$ $\displaystyle<\tilde{g}_{x}$ (5a)
$\displaystyle F_{u}(Wx_{0}+w)$ $\displaystyle<g_{u}$ (5b)

for all $x_{0}\in\mathcal{S}$, then $\pi_{0}(x_{0})=Wx_{0}+w$ solves (4). The
following optimization problem can be solved to find $W$ and $w$ or certify
that none exists. Let $\mathcal{Y}(s)=\\{x_{0}\in\mathbb{R}^{n}\mid
F_{s}(Ax_{0}+B(Wx_{0}+w))\leq\tilde{g}_{s}+s\textbf{1},F_{u}(Wx_{0}+w)\leq
g_{u}+s\textbf{1}\\}.$ If the optimal cost of

$\displaystyle\min_{W,w,s}s\text{ subject to
}{\color[rgb]{0,0,0}\mathcal{S}}\subseteq\mathcal{Y}(s)$ (6)

is negative, then (5) holds for all $x_{0}\in\mathcal{S}$, thus $\pi_{0}$
solves (4). This happens to be the case for the example in Section V, taken
from [6]. The constraint in (6) is a polytope containment constraint in
halfspace representation, thus (6) can be solved as a linear program [28].

Now consider the feasibility problem for $\mathcal{F}(x_{0})$, which is
obtained by replacing (4b) and (4c) with (3a) and (3b), and changing the
optimization variable from $u\in\mathbb{R}^{m}$ to
$\textbf{u}\in\mathbb{R}^{m\tau}$. One would naturally expect the complexity
of the PWA solution to this feasibility problem to increase rapidly with the
time horizon $\tau$, as more decision variables and constraints are added.
However, the next proposition shows that the cardinality of the stored
partition can be made constant in $\tau$.

######  Proposition 1 (Phase I solution)

If $\pi_{0}$ solves (4), then the vector
$\mu_{0}(x_{0}):=\begin{bmatrix}\pi_{0}(x_{0})^{T},\ldots,\pi_{0}(x_{\tau-1})^{T}\end{bmatrix}^{T}$,
where $x_{k+1}=Ax_{k}+B\pi_{0}(x_{k})$, is an interior point of
$\mathcal{F}(x_{0})$ for any $x_{0}\in$ $\mathcal{S}$.

###### Proof:

If $\pi_{0}$ solves (4), then $\pi_{0}(x)\in\textbf{int }\mathcal{V}(x)$ for
all $x\in\mathcal{S}$. Applying the definition of $\mathcal{V}$ in an
inductive argument, it is straightforward to show that the state trajectory
associated with $\mu_{0}(x_{0})$ is entirely contained in $\mathcal{S}$. Fix
any such trajectory $\\{x_{1},\ldots,x_{\tau}\\}\subset\mathcal{S}$
originating from $x_{0}\in\mathcal{S}$ under policy $\pi_{0}$. For any
$k\in\\{1,\ldots,\tau\\}$, $x_{k}\in\mathcal{S}$ implies
$\pi_{0}(x_{k})\in\textbf{int }\mathcal{V}(x_{k})$, which implies
$\pi_{0}(x_{k})\in\textbf{int }\mathcal{U}$ and
$Ax_{k}+B\pi_{0}(x_{k})\in\textbf{int }\mathcal{T}$. Since this holds for all
$k$, the constraints defining $\mathcal{F}(x_{0})$ hold strictly at
$\mu_{0}(x_{0})$. ∎

In our simulations on the example from [6], $\eqref{eqn:3-17-4}$ was feasible
with negative optimal cost, meaning that a polyhedral partition of the state
space was not needed (see Section V). This indicates that the minimum number
of regions in a polyhedral state space partition associated with a PWA
solution to (4) is in general very small relative to the number of regions in
an explicit solution to (2).

##  IV Phase II: Optimizing Performance

In this section, we construct a class of policies from $x_{0}\in\mathcal{S}$
to $\mathcal{F}(x_{0})$, that can be trained using standard machine learning
packages. Although it is difficult to constrain the output of a neural network
to an arbitrary polytope such as $\mathcal{F}(x_{0})$, it is easy to constrain
the output to the hypercube $\mathbb{B}_{\infty}$ by applying a clamping
function elementwise in the output layer. We apply a mapping between polytopes
that is closed-form, differentiable, and bijective. This mapping establishes
an equivalence between $\mathbb{B}_{\infty}$ and $\mathcal{F}(x_{0})$,
allowing one to constrain the outputs of the policy to $\mathcal{F}(x_{0})$.
The mapping from $\mathbb{B}_{\infty}$ to $\mathcal{F}(x_{0})$ is called the
gauge map. The concept is illustrated in Figure 2.

Figure 2: The proposed control policy uses a neural network combined with the
Phase I solution and a gauge map to constrain the decision u to the MPC
feasible set $\mathcal{F}(x_{0})$. The first action from the sequence u is
extracted and implemented. On the right, the action of the gauge map is
illustrated.

We begin constructing the gauge map by introducing some preliminary concepts.
A C-set is a convex, compact set that contains the origin as an interior
point. The gauge function with respect to C-set
$\mathcal{P}\subset\mathbb{R}^{n}$, denoted
$\gamma_{\mathcal{P}}:\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}$, is the
function whose sublevel sets are scaled versions of $\mathcal{P}$.
Specifically, the gauge of a vector $v$ with respect to $\mathcal{P}$ is given
by $\gamma_{\mathcal{P}}(v)=\inf\\{\lambda\geq 0\mid
v\in\lambda\mathcal{P}\\}.$ If $\mathcal{P}$ is a polytopic C-set given by
$\\{v\in\mathbb{R}^{k}\mid Fv\leq g\\}$, then $\gamma_{\mathcal{P}}$ is the
pointwise maximum over a finite set of affine functions [17]:

$\displaystyle\gamma_{\mathcal{P}}(v)=\max_{i}\frac{F^{(i)T}v}{g^{(i)}}.$ (7)

Given two C-sets $\mathcal{P}$ and $\mathcal{Q}$, the gauge map
$G:\mathcal{P}\rightarrow\mathcal{Q}$ is

$\displaystyle
G(v\mid\mathcal{P},\mathcal{Q})=\frac{\gamma_{\mathcal{P}}(v)}{\gamma_{\mathcal{Q}}(v)}\cdot
v.$ (8)

This function maps level sets of $\gamma_{\mathcal{P}}$ to level sets of
$\gamma_{\mathcal{Q}}$.

###### Proposition 2

Given two polytopic C-sets $\mathcal{P}$ and $\mathcal{Q}$, the gauge map
$G:\mathcal{P}\rightarrow\mathcal{Q}$ is subdifferentiable and bijective.
Further, given a function $\pi_{0}$ from Proposition 1, the set
$\tilde{\mathcal{F}}(x):=[\mathcal{F}(x)-\pi_{0}(x)]$ is a C-set for all
$x\in\mathcal{S}$.

###### Proof:

The properties of subdifferentiability and bijectivity come from [17]. For the
C-set property, fix $x\in\mathcal{S}$. Since $\mathcal{S},$ $\mathcal{U},$ and
$\mathcal{D}$ are convex and compact, so is $\mathcal{F}(x)$. Since
$\mu_{0}(x)$ is an interior point of $\mathcal{F}(x),$ the set
$\tilde{\mathcal{F}}(x)$ contains the origin as an interior point and is
therefore a C-set. ∎

We now use the gauge map in conjunction with the Phase I solution to construct
a neural network whose output is confined to $\mathcal{F}(x_{0}).$ Let
$\psi_{\theta}:\mathcal{S}\rightarrow\mathbb{B}_{\infty}$ be a neural network
parameterized by $\theta$. A safe policy is constructed by composing the gauge
map $G:\mathbb{B}_{\infty}\rightarrow\tilde{\mathcal{F}}(x_{0})$ with
$\psi_{\theta}$, then adding $\mu_{0}(x_{0})$ to map the solution into
$\mathcal{F}(x_{0})$:

$\displaystyle\mu_{\theta}(x_{0})=G(\cdot\mid\mathbb{B}_{\infty},\tilde{\mathcal{F}}(x_{0}))\circ\psi_{\theta}(x_{0})+\mu_{0}(x_{0}).$
(9)

Computing the gauge map online simply requires evaluating $H_{s}M_{0}x_{0}$
from (3a) as well as the operations in (7).

The function $\mu_{\theta}$ has several important properties for approximating
the optimal solution to (2). First, it leverages the universal function
approximation properties of neural networks [29] along with the bijectivity of
the gauge map (Proposition 2) to explore all interior points of
$\mathcal{F}(x_{0}).$ This is an advantage over projection-based methods [11]
which may be biased towards the boundary of $\mathcal{F}(x_{0})$ when the
optimal solution may lie on the interior. Second, $\mu_{\theta}$ is evaluated
in closed form, and its outputs are constrained to $\mathcal{F}(x_{0})$
without the use of an optimization layer [14] that may have high computational
overhead. Finally, the subdifferentiability of the gauge map (Proposition 2)
enables selection of parameter $\theta$ using standard automatic
differentiation techniques.

### Optimizing the parameter $\theta$

Similar to the approach taken in [10], we optimize $\theta$ by sampling
$x\in\mathcal{S}$ and applying stochastic gradient descent. At each iteration,
a new batch of initial conditions $\\{x_{0}^{j}\\}_{j=1}^{M}$ is sampled from
$\mathcal{S}$ and the loss is computed as

$\displaystyle
J(\theta)=\frac{1}{M}\sum_{j=1}^{M}\sum_{k=0}^{\tau-1}l(x_{k}^{j},u_{k}^{j})+l_{F}(x_{\tau}^{j})$
(10)

with the control sequences $\textbf{u}^{j}$ given by $\mu_{\theta}(x_{0}^{j})$
and state trajectories $\textbf{x}^{j}$ generated according to the nominal
dynamics. The parameters $\theta$ are updated in the direction of
$\nabla_{\theta}J$, which is easily computed using automatic differentiation
[30].

##  V Simulations

###  V-A Test systems

We simulate the proposed policy using a modified example from [6] with $n=3$,
$m=2,$ and $\tau=5$. The system matrices, constraints, costs, and Phase I
solution (found using (6)) are given below:

$\displaystyle A=\begin{bmatrix}-.5&.3&-1\\\ .2&-.5&.6\\\
1&.6&-.6\end{bmatrix},\ B=\begin{bmatrix}-.601&-.890\\\ .955&-.715\\\
.246&-.184\end{bmatrix},$ (11) $\displaystyle\|x\|_{\infty}\leq 5,\
\|u\|_{\infty}\leq 1,\ \|d\|_{\infty}\leq 0.1,$ (12) $\displaystyle
l(x,u)=\|x\|_{2}^{2}+c_{1}\|u\|_{2}^{2},\ l_{F}(x)=c_{2}\|x\|_{2}^{2}$ (13)
$\displaystyle W=\begin{bmatrix}0.116&0.210&-0.370\\\
-0.320&-0.104&-0.122\end{bmatrix},w=\begin{bmatrix}-0.157\\\
-0.0533\end{bmatrix}$

where $c_{1}$ and $c_{2}$ are positive constants. Although quadratic costs are
used in the simulations, the proposed method can work with any differentiable
cost.

We evaluate the performance of a given policy in both open- and closed-loop
experiments. In the open-loop experiments, we evaluate the MPC cost (2a) and
compare it to the optimal cost. The fraction suboptimality is

$\displaystyle\delta=\frac{c_{nn}-c_{mpc}}{c_{mpc}}$ (14)

where $c_{nn}$ is the average cost (2a) incurred by the control sequence
$\mu_{\theta}$ on a validation set
$\\{x_{0}^{j}\\}_{j=1}^{N_{val}}\subset\mathcal{S}$ and $c_{mpc}$ is the
optimal cost.

In the closed-loop experiments, we evaluate the performance of a policy
$\pi_{\theta}(x_{t}):\mathbb{R}^{n}\rightarrow\mathbb{R}^{m},t\geq 0$ which is
derived from $\mu_{\theta}(x_{t})$ by taking the first action in the sequence.
We simulate (1) for $T\gg\tau$ time steps. The trajectory cost in the closed-
loop experiments is computed as $\sum_{t=0}^{T-1}l(x_{t},u_{t})+l_{F}(x_{T})$
and the disturbance is modeled as an autoregressive sequence [31],
$d_{t+1}=\alpha d_{t}+(1-\alpha)\hat{d}$ where $\alpha\in(0,1)$ and $\hat{d}$
is drawn uniformly over $\mathcal{D}$.

###  V-B Benchmarks

We compare the proposed method to two of the most common approaches for
learning a solution to (2). The first benchmark is a penalty-based approach
[32] which enforces the constraints (2c) and (2d) by augmenting the cost (10)
with a linear penalty term on constraint violations given by
$\beta\cdot\max\\{0,F_{x}x_{t}-\tilde{g}_{x}\\}$ where the $\max$ is evaluated
elementwise and $\beta>0$. Since the penalty-based approach does not encode
state constraints in the policy, the policy is constrained to the Cartesian
product $\mathcal{U}^{\tau}=\prod_{k=0}^{\tau-1}\mathcal{U}$ using scaled
$\tanh$ functions elementwise.

The second benchmark is a projection-based approach [11] which constrains the
policy to the set $\mathcal{F}(x_{0})$ by solving a convex quadratic program
in the output layer of a neural network [33]. The optimization layer
$\textbf{v}\rightarrow\textbf{u}$ returns

$\displaystyle\underset{\textbf{u}}{\arg\min}\|\textbf{v}-\textbf{u}\|_{2}^{2}\text{
subject to }\textbf{u}\in\mathcal{F}(x_{0}).$

Another class of approaches to learning-based MPC seeks to learn the optimal
solution to (2) using regression [12, 13, 14]. Specifically, data-label pairs
$(x_{0},u_{0}^{*})$ are generated by sampling $x_{0}$ from $\mathcal{S}$,
solving (2) for each sample, and extracting $u_{0}^{*}$ from the optimal
solution $\textbf{u}^{*}$. Then, a neural network or other function
approximator is trained to learn the relationship between $x_{0}$ and
$u_{0}^{*}$. Performance and constraint satisfaction are handled e.g. by
bounding the approximation error with respect to the MPC oracle. We do not
compare against this type of approach since it requires a large number of
trained samples, making it difficult to compare with our and the other
unsupervised examples.

###  V-C Neural network design

The neural networks were designed with $n$ inputs, $m\tau$ outputs, and two
hidden layers with rectified linear unit (ReLU) activation functions. The
width of the networks was chosen during hyperparameter tuning. In particular,
we performed 30 iterations of random search over the width of the network
(number of neurons per hidden layer) $\in\\{64,\ldots,1024\\}$, the batch size
(number of initial conditions, $M$) $\in\\{100,\ldots,3000\\}$ and the
learning rate (LR, the step size for gradient descent) $\in[10^{-5},10^{-3}]$.
For each set of hyperparameters under consideration, we computed the
validation score using (14) with $N_{val}=100$. The hyperparameters after
tuning are reported in Table I.

TABLE I: Hyperparameters for the three neural networks. Type | Width | LR | $M$
---|---|---|---
Gauge | $859$ | $4.7\times 10^{-4}$ | $1655$
Penalty | $318$ | $8.7\times 10^{-4}$ | 133
Projection | $956$ | $9.0\times 10^{-5}$ | $813$

###  V-D Simulation results

Here we compare our proposed approach (Gauge NN), the penalty-based approach
(Penalty NN), the projection-based approach (Projection NN) and the “ground
truth” obtained by solving (2) online in cvxpy. The results of the open-loop
experiments are shown in Table II, with performance computed relative to the
optimal MPC solution using (14) with $N_{val}=100$ trials. The proposed Gauge
NN achieves lower cost compared to the projection-based method, and has a much
lower computational complexity (solve time is only 6% of projection). Table II
only compares the NNs with safety guarantees because constraint violations are
not accounted for in (14).

TABLE II: Open-loop test results. Type |  $\delta$ (14) | Solve time (sec)
---|---|---
Gauge | 0.007 | .0015
Projection | 0.010 | .024

Figure 3 shows the training curves for each type of network. The lower
training cost achieved by the Gauge NN illustrates that it can be more
efficient to explore the interior of the feasible set than the boundary. Since
the MPC cost in the simulations is strictly convex, solutions with lower cost
are closer to the optimal solution.

Figure 3: Training trajectories for the three types of neural netwokrs. Our
proposed Gauge-based approach achieves lower cost at a much faster rate.

Figure 4 compares the policies in terms of computation time and test
performance. The box-and-whisker plots indicate the range of performance over
100 test trajectories of length $T=50$, while the vertical position of each
box indicates the average time to compute a control action. Of the policies
with safety guarantees (Gauge NN, Projection NN, and online MPC), the Gauge NN
achieves Pareto efficiency in terms of average solve time and median
trajectory cost. Our intuition behind the high performance of the neural
networks is that (2) is a heuristic and the unsupervised learning approach can
lead to better closed-loop policies.

Figure 4: Solve time vs. trajectory cost for the networks under consideration
applied to the 3-state system. The Gauge NN is Pareto-efficient in terms of
cost and computation time compared to the other techniques with safety
guarantees (Online MPC and Projection NN).

##  VI Conclusion

In this paper, we provided an efficient way of exploring the interior of the
MPC feasible set for learning-based approximate explicit MPC, and demonstrated
the performance and computational gains that can be achieved by approaching
the problem from the interior. The paradigm relies on a Phase I solution that
exploits the structure of the MPC problem and a Phase II solution that
features a projection-free feasibility guarantee. The results compare
favorably against common approaches that use unsupervised learning, as well as
against the oracle itself used in supervised approaches. Future work includes
applications to MPC problems with convex but non-polytopic constraint sets,
and to distributed settings.

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