# Solving Fuzzy Quadratic Programming Problems with a Proposed Algorithm

Sajal Chakroborty
Department of Mathematics and Statistics
Texas Tech University
Lubbock, Texas, USA
<EMAIL_ADDRESS>
Other affiliation : East West University, Dhaka, Bangladesh, email:
<EMAIL_ADDRESS>
###### Abstract

The theory of fuzzy mathematics has been proven very effective for defining
and solving optimization problems. Fuzzy quadratic programming (FQP) is a
consequence of this approach. In this paper, an algorithm has been proposed to
solve FQP with coefficients as triangular fuzzy numbers (TFN). The proposed
algorithm converts FQP into two parametric quadratic programming (QP)
problems. These QP solutions provide a lower and upper bound on the objective
function of FQP. When these two values coincide, an optimal solution is
achieved. This algorithm has been analyzed using a numerical example and
compared with existing methods.

_Keywords_ Fuzzy Quadratic Programming, Triangular Fuzzy Number, $\alpha-$cut

##  1 Introduction

Quadratic programming (QP) is an optimization technique and a generalization
of linear programming (LP). It’s theoretical foundation was developed by M.
Frank and P. Wolfe [1]. QP is widely used in regression analysis, portfolio
optimization, image and signal processing, etc. It is a nonlinear optimization
problem that has a quadratic objective function with linear constraints. It
can be formulated in a general form as [3],

$\begin{split}\text{min}\;z=\sum_{j=1}^{n}c_{j}x_{j}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}q_{ij}x_{i}x_{j}\\\
\text{s.t.}\;\;\;\;\;\sum_{j=1}^{n}a_{ij}x_{j}\leq b_{i},\;i=1,\dots,n\\\
x_{j}\geq 0,\;j=1,\dots,n\end{split}$ (1)

where $x_{i}$ are decision variables to be determined, $c_{i}$ represent cost
coefficients, and $q_{ij}$ are the coefficients of quadratic form for
$i,j=1,2,\dots,n$. Note that the quadratic matrix $Q=\left[q_{ij}\right]$ is
symmetric and positive definite. The constraint coefficients and right hand
constants are denoted by $a_{ij}$ and $b_{i}$ respectively.
The above formulation does not take into account the uncertain and vague data.
But it can be dealt with using fuzzy sets and logic [10]. For this reason,
fuzzy optimization techniques have become very useful. Zimmermann first
applied this theory to mathematical programming [11]. Recently, many ideas
have been proposed to formulate and solve QP problems in a fuzzy environment.
For instance, Shiang-Tai Liu formulated QP as an FQP where all the
coefficients are triangular fuzzy numbers (TFN) [5]. Liu has developed an
algorithm where he proposed to create a pair of two-level mathematical
programs to calculate the upper and lower bound of the objective function of
FQP [4, 5]. Then the optimal solution was derived by solving two QP. Amin
Mansoori, Sohrab Effati, and Mohammad Eshangnezhad designed a one-layer
structured neural network model for solving the FQP [6]. They changed the FQP
to a bi-objective problem, then reduced it to a weighting problem, and finally
constructed a Lagrangian dual. They have solved their proposed dynamical
system by ODE method. To plan a waste management system under uncertainty, Y.
P. Li and G. H. Huang developed fuzzy two-stage quadratic programming (FTSQP)
[2]. Q. Tan, G. H. Huang, and Y. P. Cai developed a superiority-inferiority-
based inexact fuzzy stochastic quadratic programming (SI-IFSQP) model for
sustainable water supply under multiple uncertainties [7].
Motivated by the existing and recent developments in fuzzy optimization, a new
algorithm has been developed in this paper to solve FQP. This paper has been
organized as, in section 2, the definitions and properties of fuzzy sets have
been discussed briefly. In section 3, a new approach has been proposed to
solve FQP by converting it into two QP problems. The proposed method will be
demonstrated using a numerical example in the next section. Then a comparison
has been presented with existing methods. Finally, a conclusion has been made.

##  2 Preliminaries

A few definitions and properties of fuzzy set theory and optimization are
briefly explored in this section [8, 9, 11].

######  Definition 2.1.

Let $\mathcal{X}$ be a universal set. Then a fuzzy set $\tilde{A}$ in
$\mathcal{X}$ with the membership function
$\mu_{\tilde{A}}:\mathbb{R}\rightarrow[0,1]$ is defined as,

$\tilde{A}=\\{(x,\mu_{\tilde{A}}(x)):x\in\mathcal{X}\\}$

Support of $\tilde{A}$ is a crisp set
$C(\tilde{A})=\\{x\in\mathcal{X}:\mu_{\tilde{A}}(x)>0\\}$.

######  Definition 2.2.

The $\alpha-$level set of $\tilde{A}$ is a crisp set $A_{\alpha}$, defined as

$A_{\alpha}=\\{x\in\mathcal{X}:\mu_{\tilde{A}}(x)\geq\alpha\\},\alpha\in\left[0,1\right]$

######  Definition 2.3.

$\tilde{A}$ is called convex if $\mu_{\tilde{A}}\left(\lambda
x_{1}+(1-\lambda)x_{2}\right)\geq
min\\{\mu_{\tilde{A}}(x_{1}),\mu_{\tilde{A}}(x_{2})\\}$, where
$x_{1},x_{2}\in\mathcal{X}$, $\lambda\in\left[0,1\right]$.

######  Definition 2.4.

If $\mu_{\tilde{A}}=1$ for at least one $x\in\mathcal{X}$ then $\tilde{A}$ is
called normal.

######  Definition 2.5.

If $\tilde{A}$ is a normal, and convex fuzzy set with a bounded support then
it is called a fuzzy number (FN).

######  Definition 2.6.

The Triangular fuzzy number (TFN) $\tilde{A}=\left(a_{1},a_{2},a_{3}\right)$
is a FN with the membership function defined as

$\mu_{\tilde{A}}(x)=\begin{cases}\frac{x-a_{2}}{a_{2}-a_{1}},&x\in\left[a_{1},a_{2}\right]\\\
\frac{a_{3}-x}{a_{3}-a_{2}},&x\in\left[a_{2},a_{3}\right]\\\
0,&x<a_{1}\;and\;x>a_{3}\end{cases}$ (2)

where $a_{1},a_{2},a_{3}\in\mathbb{R}$ and $a_{1}\leq a_{2}\leq a_{3}$.

Note that a FN can be fully and uniquely represented by its $\alpha-$cut. The
$\alpha-$cut for a TFN is
$\tilde{A}_{\alpha}=\left[A_{\alpha}^{L},A_{\alpha}^{U}\right]$,
$\forall\alpha\in\left[0,1\right]$, where
$A_{\alpha}^{L}=a_{1}+\alpha(a_{2}-a_{1})$, and
$A_{\alpha}^{U}=a_{3}-\alpha(a_{3}-a_{2})$. This representation allow us to
perform arithmetic operation between two FN $\tilde{A}$, and $\tilde{B}$ as,

  1. i

Addition:
$(\tilde{A}+\tilde{B})_{\alpha}=[A^{L}_{\alpha}+B^{L}_{\alpha},A^{U}_{\alpha}+B^{U}_{\alpha}]$

  2. ii

Scalar Multiplication: If $k>0$ then
$(k\tilde{A})_{\alpha}=[kA^{L}_{\alpha},kA^{U}_{\alpha}]$ and
$(k\tilde{A})_{\alpha}=[kA^{U}_{\alpha},kA^{L}_{\alpha}]$ if $k<0$.

  3. iii

Multiplication:
$(\tilde{A}.\tilde{B})_{\alpha}=\left[M_{\min},M_{\max}\right]$, where
$M_{min}=\min(A^{L}_{\alpha}B^{L}_{\alpha},A^{L}_{\alpha}B^{U}_{\alpha},A^{U}_{\alpha}B^{L}_{\alpha},A^{U}_{\alpha}B^{U}_{\alpha})$
and
$M_{\max}=\max(A^{L}_{\alpha}B^{L}_{\alpha},A^{L}_{\alpha}B^{U}_{\alpha},A^{U}_{\alpha}B^{L}_{\alpha},A^{U}_{\alpha}B^{U}_{\alpha})$.

A subset $A$ of $X$ is said to be convex, if $\lambda
x+\left(1-\lambda\right)y\in A$ whenever $x,y\in A$ and
$\lambda\in\left(0,1\right)$. Let $E$ denotes a collection of TFN. Then,

######  Definition 2.7.

A fuzzy mapping $\tilde{f}:A\rightarrow E$ defined on a convex susbset $A$ in
$\mathcal{X}$ is convex, if and only if,

$\tilde{f}\left(\lambda
x+\left(1-\lambda\right)y\right)\leq\lambda\tilde{f}(x)+\left(1-\lambda\right)\tilde{f}(y),\forall
x,y\in A,\lambda\in\left[0,1\right]$

If $\tilde{f}\left(\lambda
x+\left(1-\lambda\right)y\right)<\lambda\tilde{f}(x)+\left(1-\lambda\right)\tilde{f}(y),\forall
x,y\in A,x\neq y,\lambda\in\left(0,1\right)$, then $\tilde{f}$ is called
strictly convex.

Like the $\alpha$-cut of a fuzzy number, a fuzzy mapping $\tilde{f}$ can be
written as follow,

$\tilde{f}(x)=\left[f^{L}_{\alpha},f^{U}_{\alpha}\right],\forall\alpha\in\left[0,1\right]$

Now a fuzzy optimization problem can be defined as,

######  Definition 2.8.

Let $\Omega\subset\mathbb{R}^{n}$ be a convex set which contains all feasible
solutions, then

$\begin{split}\min\;\tilde{f}(x)&=\left[f^{L}_{\alpha},f^{U}_{\alpha}\right]\\\
\text{s.t.}\;\;x&\in\Omega\end{split}$

In the following section, FQP has been formulated with fuzzy coefficients as
TFN.

##  3 Fuzzy Quadratic Programming

FQP is similar to QP in terms of formulation. It can be formulated as,

$\begin{split}\min\;\tilde{z}=\sum_{j=1}^{n}\tilde{c}_{j}x_{j}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\tilde{q}_{ij}x_{i}x_{j}\\\
\text{s.t.}\;\;\;\;\;\sum_{j=1}^{n}\tilde{a}_{ij}x_{j}\leq\tilde{b}_{i},i=1,\dots,n\\\
x_{j}\geq 0,\;j=1,\dots,n\end{split}$ (3)

where the coefficients $\tilde{c},\tilde{q}_{ij},\tilde{a}_{ij}$ and
$\tilde{b}$ are triangular fuzzy number (TFN). For $i,j=1,2,\dots,n$ let
$\tilde{c}_{j}=\left(c^{1}_{j},c^{2}_{j},c^{3}_{j}\right)$,
$\tilde{q}_{ij}=\left(q^{1}_{ij},q^{2}_{ij},q^{3}_{ij}\right)$,
$\tilde{a}_{ij}=\left(a^{1}_{ij},a^{2}_{ij},a^{3}_{ij}\right)$, and
$\tilde{b}_{i}=\left(b^{1}_{i},b^{2}_{i},b^{3}_{i}\right)$. In vector matrix
notation (3) can be simplified as,

$\begin{split}\min\;\tilde{Z}=\tilde{C}^{T}X&+\frac{1}{2}X^{T}\tilde{Q}X\\\
\text{s.t.}\;\;\tilde{A}X&\leq\tilde{b}\\\ X&\geq 0\end{split}$ (4)

######  Proposition 3.1.

If $\tilde{Q}$ is symmetric and positive semi-definite, then
$\tilde{f}(x)=\tilde{C}^{T}X+\frac{1}{2}X^{T}\tilde{Q}X$ in (4) is a convex
fuzzy mapping.

###### Proof.

For any $\lambda\in\left[0,1\right]$, $\lambda x+(1-\lambda)y\in\Omega$,
implies,

$\begin{split}\tilde{f}\left(x\lambda+(1-\lambda)y\right)&=\tilde{C}^{T}(x\lambda+(1-\lambda)y)+\frac{1}{2}\left(\lambda
x+(1-\lambda)y\right)^{T}\tilde{Q}\left(\lambda x+(1-\lambda)y\right)\\\
&=\tilde{C}^{T}(x\lambda+(1-\lambda)y)+\frac{1}{2}\left(y+\lambda(x-y)\right)^{T}\tilde{Q}\left(y+\lambda(x-y)\right)\\\
&=\tilde{C}^{T}(x\lambda+(1-\lambda)y)+\frac{1}{2}\left(y^{T}\tilde{Q}y+\lambda^{2}(x-y)^{T}\tilde{Q}(x-y)+2\lambda
y^{T}\tilde{Q}(x-y)\right)\\\
&\leq\tilde{C}^{T}(x\lambda+(1-\lambda)y)+\frac{1}{2}\left(y^{T}\tilde{Q}y+\lambda(x-y)^{T}\tilde{Q}(x-y)+2\lambda
y^{T}\tilde{Q}(x-y)\right)\\\
&=\tilde{C}^{T}(x\lambda+(1-\lambda)y)+\frac{1}{2}\left(y^{T}\tilde{Q}y+\lambda(x+y)^{T}\tilde{Q}(x-y)\right)\\\
&=\tilde{C}^{T}(x\lambda+(1-\lambda)y)+\frac{1}{2}\left(\lambda
x^{T}\tilde{Q}x+(1-\lambda)y^{T}\tilde{Q}y\right)\\\
&=\lambda\left(\tilde{C}^{T}x+\frac{1}{2}X^{T}\tilde{Q}x\right)+\left(1-\lambda\right)\left(\tilde{C}^{T}y+\frac{1}{2}y^{T}\tilde{Q}y\right)\\\
&=\lambda\tilde{f}(x)+\left(1-\lambda\right)\tilde{f}(y)\end{split}$

∎

Since the objective function of equation (4) is a convex fuzzy mapping and
$\\{X:X\geq 0,\tilde{A}X\leq\tilde{b}\\}$ is a convex feasible set, so (4) is
a convex fuzzy programming. Now if $X^{*}$ is a local optimal solution of (4),
then it will be a global solution of (4) [6]. Also, if the objective function
is strictly convex, the $X^{*}$ is the unique global optimal solution of (4)
according to [6]. In the next section, the proposed method has been discussed.

##  4 Proposed Method

First, define the real-life problem as (3). The arithmetic operations that
have been used here are defined in section 2. Continue with the following
steps.

Step-1 : To determine the $\alpha$-optimal value by taking the $\alpha$-cut of
the objective function and the constraints (3) can be written as,

$\begin{split}\min\;\left[z^{L}_{\alpha},z^{U}_{\alpha}\right]=\sum_{j=1}^{n}\left[c^{L}_{\alpha,j},c^{U}_{\alpha,j}\right]x_{j}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\left[q^{L}_{\alpha,ij},q^{U}_{\alpha,ij}\right]x_{i}x_{j}\\\
\text{s.t.}\;\;\;\;\;\;\sum_{j=1}^{n}\left[a^{L}_{\alpha,ij},a^{U}_{\alpha,ij}\right]x_{j}\leq[b^{L}_{\alpha,i},b^{U}_{\alpha,i}],\;i=1,\dots,n\\\
x_{j}\geq 0,\;\;j=1,\dots,n\end{split}$ (5)

Step-2 : Formulate the following QP for different $\alpha$ values to obtain
the lower bound $z^{L}_{\alpha}$ of the objective function.

$\begin{split}\min
z^{L}_{\alpha}=\sum_{j=1}^{n}c^{L}_{\alpha,j}x_{j}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}q^{L}_{\alpha,ij}x_{i}x_{j}\\\
\text{s.t.}\;\;\;\;\sum_{j=1}^{n}a^{L}_{\alpha,ij}x_{j}\leq
b^{L}_{\alpha,i},\;\;i=1,\dots,n\\\ x_{j}\geq 0,\;\;j=1,\dots,n\end{split}$
(6)

Step-3 : Similarly, to find the upper bound $z^{U}_{\alpha}$ of $\tilde{z}$
for different $\alpha$ values formulate the following QP.

$\begin{split}\min
z^{U}_{\alpha}=\sum_{j=1}^{n}c^{U}_{\alpha,j}x_{j}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}q^{U}_{\alpha,ij}x_{i}x_{j}\\\
\text{s.t.}\;\;\;\;\sum_{j=1}^{n}a^{U}_{\alpha,ij}x_{j}\leq
b^{U}_{\alpha,i},\;\;i=1,\dots,n\\\ x_{j}\geq 0,\;\;j=1,\dots,n\end{split}$
(7)

Step-4 : Choose an initial value of $x_{0}$ for the decision variables and
solve the problem (6) for a fixed $\alpha\in\left[0,1\right]$.

$\begin{split}y^{L}_{k}&=x_{k}-\frac{1}{K_{\alpha}}\nabla z^{L}_{\alpha}\\\
x_{k+1}&=proj_{\mathbb{S}_{L}}\left(y^{L}_{k}\right)\end{split}$

For fixed $\alpha$, $K_{\alpha}$ is the Lipschitz constant of $z^{L}_{\alpha}$
and $proj_{\mathbb{S}_{L}}$ is the projection on the half space
$\mathbb{S}_{L}=\\{x:A^{L}_{\alpha}x\leq b^{L}_{\alpha}\\}$.

Step-5 : Follow the same procedure as step-4 and perform the following steps
for (7).

$\begin{split}y^{U}_{k}&=x_{k}-\frac{1}{M_{\alpha}}\nabla z^{U}_{\alpha}\\\
x_{k+1}&=proj_{\mathbb{S}_{U}}\left(y^{U}_{k}\right)\end{split}$

Like previous step, $M_{\alpha}$ is the Lipschitz constant of $z^{U}_{\alpha}$
and $proj_{\mathbb{S}_{U}}$ is the projection on the half space
$\mathbb{S}_{U}=\\{x:A^{U}_{\alpha}x\leq b^{U}_{\alpha}\\}$.

Step-6 : If $z^{L}_{\alpha}=z^{U}_{\alpha}$ for any $\alpha$, then stop.

##  5 Numerical Example

In this section, effectiveness of the proposed algorithm has been analyzed
with an example. Consider the following FQP from [5]

$\begin{split}\min\;\tilde{z}=\left(-6,-5,-4\right)x_{1}+\left(1,1.5,2\right)x_{2}+\frac{1}{2}\left[\left(4,6,8\right)x_{1}^{2}+\left(-6,-4,-2\right)x_{1}x_{2}+\left(2,4,6\right)x_{2}^{2}\right]\\\
\end{split}$
$\begin{split}\text{s.t.}\;\;\;\;\;\;\;x_{1}+\left(0.5,1,1.5\right)x_{2}\leq\left(1,2,3\right)\\\
\;\;\left(1,2,3\right)x_{1}+\left(-2,-1,-0.5\right)x_{2}&\leq\left(3,4,5\right)\\\
x_{1},x_{2}\geq 0&\end{split}$

At first, two QP models will be formulated according to Step-$3$ and $4$. For
any $\alpha\in\left[0,1\right]$ the lower bound $z^{L}_{\alpha}$ of
$\tilde{z}$ will be obtained by solving,

$\begin{split}\min\;\;z^{L}_{\alpha}=\left(-6+\alpha\right)x_{1}+\left(1+0.5\alpha\right)x_{2}+\frac{1}{2}&\left(x_{1}\;x_{2}\right)\left(\begin{array}[]{cc}4+2\alpha&-3+\alpha\\\
-3+\alpha&2+2\alpha\end{array}\right)\left(\begin{array}[]{cc}x_{1}\\\
x_{2}\end{array}\right)\\\
\text{s.t.}\;\left(\begin{array}[]{cc}1&0.5+0.5\alpha\\\
1+\alpha&-2+\alpha\end{array}\right)\left(\begin{array}[]{cc}x_{1}\\\
x_{2}\end{array}\right)&\leq\left(\begin{array}[]{cc}1+\alpha\\\
2+2\alpha\end{array}\right)\\\ x_{1},x_{2}&\geq 0\end{split}$ (8)

Similarly, to get the upper bound of $\tilde{Z}$, solve the following QP.

$\begin{split}\min\;\;z^{U}_{\alpha}=\left(-4-\alpha\right)x_{1}+\left(2-0.5\alpha\right)x_{2}+&\frac{1}{2}\left(x_{1}\;x_{2}\right)\left(\begin{array}[]{cc}8-2\alpha&-1-\alpha\\\
-1-\alpha&6-2\alpha\end{array}\right)\left(\begin{array}[]{cc}x_{1}\\\
x_{2}\end{array}\right)\\\
\text{s.t.}\;\;\left(\begin{array}[]{cc}1&1.5-0.5\alpha\\\
3-\alpha&0.5-0.5\alpha\end{array}\right)&\left(\begin{array}[]{cc}x_{1}\\\
x_{2}\end{array}\right)\leq\left(\begin{array}[]{cc}3-\alpha\\\
6-2\alpha\end{array}\right)\\\ x_{1},x_{2}&\geq 0\end{split}$ (9)

The detailed results with comparison to existing methods have been discussed
in the following section.

##  6 Results and Discussion

For different values of $\alpha$, the objective function values of (8) and (9)
have been presented in Table 1. Solutions obtained by [5] and [6] are also
presented in the same table with computation time. In Figure 1, the membership
function of $\tilde{z}$ obtained by the proposed method has been presented
along with Liu and Mansoori’s method. In Figure 2, a comparison between the
proposed method and the existing methods is shown.

Table 1: Objective values for different $\alpha$
$\alpha$ | 0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |  | CPU
---
time (s)
|  $z^{L}_{\alpha}$ in Proposed
---
Method
-4.0833 | -4.0503 | -3.7271 | -3.1306 | -2.4891 | -2.0872 | 0.032
|  $z^{U}_{\alpha}$ in Proposed
---
Method
-1 | -1.1605 | -1.3444 | -1.5559 | -1.8 | -2.0872 | 0.029
|  $z^{L}_{\alpha}$ in Liu’s [5]
---
Method
-10.08 | -6.72 | -4.46 | -3.14 | -2.49 | -2.09 | 0.050
|  $z^{U}_{\alpha}$ in Liu’s [5]
---
Method
-1 | -1.16 | -1.34 | -1.45 | -1.80 | -2.09 | 0.055
|  $z^{L}_{\alpha}$ in Mansoori’s
---
Method [6] with
$w_{1}=\frac{1}{4}$, $w_{2}=\frac{3}{4}$
-1.4464 | -1.559 | -1.6735 | -1.8 | -1.9363 | -2.0875 | 0.033
|  $z^{U}_{\alpha}$ in Mansoori’s
---
Method [6]with
$w_{1}=\frac{1}{3}$, $w_{2}=\frac{2}{3}$
-1.6333 | -1.7146 | -1.8 | -1.8997 | -1.9841 | -2.0875 | 0.034

From the above table, it is clear that the sequence of objective values
obtained by the proposed algorithm has converged at the same point as the
existing methods. It generates more accurate solutions than others. In the
following figure, the convergence of objective values for different methods is
shown in terms of the triangular membership function.

Figure 1: Membership function for the Proposed Method vs Existing Methods

The gap between the lower and upper bounds of the objective values of Liu’s
method is larger than that of the proposed method. Although this gap is larger
than Mansoori’s method, anyone can make an intuition that the proposed method
is slower than Mansoori’s method. But the computation time tells another
story. Though it could be easily observed from Table 1, let’s look at the
following figure.

Figure 2: Computational Time Comparison

The proposed algorithm has taken 0.0305 seconds on average to solve the
problem. On the other hand, Mansoori’s method took 0.0335 seconds and Liu’s
method took 0.0525 seconds on average.

##  7 Conclusion

Solving FQP is more challenging than QP. In this paper, a new approach has
been proposed to solve FQP. This approach has generated more accurate results
than many existing methods while using less computation time. This idea can be
easily extended to solve multi-objective FQP, fuzzy regression analysis, etc.,
which is the future goal of this work.

## References

  * [1] Marguerite Frank and Philip Wolfe.  An algorithm for quadratic programming.  Naval research logistics quarterly, 3(1-2):95–110, 1956.
  * [2] YP Li and Guo H Huang.  Fuzzy two-stage quadratic programming for planning solid waste management under uncertainty.  International Journal of Systems Science, 38(3):219–233, 2007.
  * [3] Hiller Liberman.  Introduction to operations research.  Libros McGraw-Hill de Mexico, 1988.
  * [4] Shiang-Tai Liu.  Solving quadratic programming with fuzzy parameters based on extension principle.  In 2007 IEEE International Fuzzy Systems Conference, pages 1–5. IEEE, 2007.
  * [5] Shiang-Tai Liu.  A revisit to quadratic programming with fuzzy parameters.  Chaos, Solitons & Fractals, 41(3):1401–1407, 2009.
  * [6] Amin Mansoori, Sohrab Effati, and Mohammad Eshaghnezhad.  A neural network to solve quadratic programming problems with fuzzy parameters.  Fuzzy Optimization and Decision Making, 17(1):75–101, 2018.
  * [7] Qian Tan, GH Huang, and YP Cai.  Multi-source multi-sector sustainable water supply under multiple uncertainties: an inexact fuzzy-stochastic quadratic programming approach.  Water resources management, 27(2):451–473, 2013.
  * [8] Guixiang Wang and Congxin Wu.  Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming.  Fuzzy Sets and Systems, 138(3):559–591, 2003.
  * [9] Hsien-Chung Wu.  Evaluate fuzzy optimization problems based on biobjective programming problems.  Computers & Mathematics with Applications, 47(6-7):893–902, 2004\.
  * [10] Lotfi A Zadeh.  Fuzzy sets.  In Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh, pages 394–432. World Scientific, 1996.
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††thanks: Electronic mail<EMAIL_ADDRESS>
#  Tunneling time, the barrier time-delay and weak measurement
What was or can actually be measured with the Attoclock?

Ossama Kullie  Theoretical Physics, Institute for Physics, Department of
Mathematics and Natural Science, University of Kassel, Germany

###### Abstract

The measurement of the tunneling time-delay in attosecond experiments, termed
attoclock, despite its controversies, offers a fruitful opportunity to
understand the tunneling time and the tunnel-ionization process. In previous
work, we showed a model that well describes the time-delay measured by the
attoclock experiment in the adiabatic and nonadiabatic field calibrations. In
the present work we show that the tunneling time reveals a universal behavior,
with disentangled contribution, where the barrier time-delay can be
convincingly defined and determined from the difference between the time-delay
of adiabatic and nonadiabatic tunnel-ionization, with good agreement with the
experimental result. Furthermore, we show that in the weak measurement limit,
the barrier time-delay corresponds to the Larmor-clock time and the
interaction time within the barrier.

Ultrafast science, attosecond physics, tunneling and tunnel-ionization time-
delay, nonadiabatic effects, weak measurement, Lamor clock, interaction time,
time and time-operator in quantum mechanics.

##  .1 Theoretical model

The interaction of an atom with laser pulse in the strong-field and attosecond
science can be modeled in a simplified picture as shown in fig. 1. With its
help, we developed a simple tunneling model in Kullie (2015) with expressions
composed of basic laser and atomic parameters that describes the measurement
result of the attocolck. The model provides a new insights into temporal
property of tunneling ionization, and shed light on roles of time in quantum
mechanics. Below we briefly present our model, in which an electron can be
tunnel-ionized by a laser pulse with an electric field strength (hereafter
field strength) $F$. A direct ionization happen when the field strength
reaches a threshold called atomic field strength $F_{a}=I_{p}^{2}/(4Z_{eff})$
Augst  _et al._ (1989, 1991), where $I_{p}$ is the ionization potential of the
system (atom or molecule) and $Z_{eff}$ is the effective nuclear charge in the
single-active electron approximation (SAEA). We adopt in this work the atomic
units ($au$), where the mass and charge of the electron as well as the Planck
constant set to unity, $\hbar=m=e=1$. However, for $F<F_{a}$ the ionization
can happen by tunneling through a potential barrier of an effective potential
consisting of the Coulomb potential of the core and the electric field of the
laser pulse. It can be expressed in a one-dimensional form by

$V_{eff}(x)=V(x)-xF=-\frac{Z_{eff}}{x}-xF,$ (1)

compare fig. 1. In the model the tunneling process can be described solely by
the ionization potential $I_{p}$ of the valence (the interacting) electron and
the peak field strength $F$ with the barrier height is given by
$\delta_{z}=\sqrt{I_{p}^{2}-4Z_{eff}F}$, where $F$ stands (throughout this
work) for the peak electric field strength at maximum. In fig. 1 (for details
see Kullie (2015)), the inner (entrance $x_{e,-}$) and outer (exit $x_{e,+}$)
points are given by $x_{e,\pm}=(I_{p}\pm\delta_{z})/(2F)$, the barrier width
is $d_{B}=x_{e,+}-x_{e,-}=\delta_{z}/F$, and its (maximum) height (at
$x_{m}(F)=\sqrt{Z_{eff}/F}$) is $\delta_{z}$. At $F=F_{a}$, we have
$\delta_{z}=0$ ($d_{B}=0$), the barrier disappears and the direct or the
barrier-suppression ionization starts, green (dashed-doted) curve in fig.1 .

Figure 1: (Color online) A sketch of the atomic and the effective potential
curves, showing the height and width of the barrier formed by the laser field.
The plot is for He-atom in the SAEA model with $Z_{eff}=1.6875$ and
$I_{p}\approx 0.9\,au$.

### Adiabatic tunneling:

In the adiabatic field calibration of the attoclock of Landsman et. al.
Landsman  _et al._ (2014), we showed in Kullie (2015) that the tunneling time-
delay is expressed by the forms,

$\displaystyle\tau_{{}_{T,d}}=\frac{1}{2(Ip-\delta_{z})},\quad\tau_{{}_{T,i}}=\frac{1}{2(I_{p}+\delta_{z})}$
(2)

It was shown Kullie (2015) that $\tau_{{}_{T,d}}$ agrees well with the
experimental result of Landsman et. al. Landsman  _et al._ (2014) . Their
physical reasoning is the following: $\tau_{{}_{T,d}}$ is the time-delay of
the adiabatic tunnel-ionization with respect to the ionization at atomic field
strength $F_{a}$ required to overcome the barrier and escape at the exit point
$x_{e,+}$ into the continuum Kullie (2015). Whereas $\tau_{{}_{T,i}}$ is the
time needed to reach the entrance point $x_{{}_{e,-}}$ from the initial point
$x_{i}$, compare fig. 1. At the limit of atomic field strength, we have
$\lim\limits_{F\to F_{a}}(\delta_{z}\to
0)\tau_{{}_{T,d}}=\frac{1}{2Ip}=\tau_{a}$ (see below). For $F>F_{a}$, the
barrier-suppression ionization sets up Delone and Kraǐnov (1998); Kiyan and
Kraǐnov (1991). On the opposite side, we have $\lim\limits_{F\to
0}(\delta_{z}\to I_{p})\tau_{{}_{T,d}}\to\infty$. So nothing happens and the
electron remains in its ground state undisturbed, which shows that our model
is consistent. For details, see Kullie (2015, 2016, 2018, 2020).

### Nonadiabatic tunneling:

In the nonadiabatic field calibration of Hofmann et. al. Hofmann  _et al._
(2019), we found in Kullie (2024) that the time-delay of the nonadiabatic
tunnel-ionization is descried by

$\displaystyle\tau_{dion}(F)$ $\displaystyle=$
$\displaystyle\frac{1}{2I_{p}}\frac{F_{a}}{F}=\tau_{a}\xi_{F}$ (3)

In Kullie (2024) it was shown that 3 agrees well with the experimental result
of Hofmann et. al. Hofmann  _et al._ (2019). In addition, the result was
confirmed by the numerical integration of the time-dependent Schrödinger
equation (NITDSE) Kullie (2024). It is easily seen that in the limit $F\to
F_{a},\tau_{dion}=\tau_{a}$, whereas in the opposite case $F\to
0,\tau_{dion}=\infty$. $\tau_{a}$ is always (adiabatic and nonadiabatic) a
lower quantum limit of the tunnel-ionization time-delay and quantum
mechanically does not vanish. $\xi(F)$ is an enhancement factor for field
strength $F<F_{a}$.

##  .2 The barrier time-delay

In this section, we show that the time-delay due to the barrier itself, which
is the actual tunneling time-delay can be determined by considering the
adiabatic and nonadiabatic field calibrations together. Eq 2 can be decomposed
into a twofold time-delay with respect to ionization at $F_{a}$, representing
$\tau_{{}_{T,d}}$ in an unfolded form,

$\displaystyle\tau_{{}_{T,d}}$ $\displaystyle=$
$\displaystyle\frac{1}{2(Ip-\delta_{z})}=\frac{1}{2}\frac{I_{p}}{4Z_{eff}F}\left(1+\frac{\delta_{z}}{I_{p}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2I_{p}}\frac{F_{a}}{F}\left(1+\frac{\delta_{z}}{I_{p}}\right)=\tau_{a}(\xi_{F}(F)+\Lambda_{F}(F))$
$\displaystyle=$ $\displaystyle\tau_{dion}+\tau_{delt}\equiv\tau_{Ad}$

Below we refer to it as adiabatic tunnel-ionization $\tau_{Ad}$. Eq. .2
immediately shows in the second line that the adiabatic time-delay can be
easily interpreted as a time-delay with respect to ionization at atomic field
strength $\tau_{a}=\tau_{{Ad}}(F_{a})=1/(2I_{p})$, where both terms present
real time-delay. And again $(\xi_{F}+\Lambda_{F})$ is an enhancement factor
for field strength $F<F_{a}$. The first term of $\tau_{dion}$ eq. .2, which is
the same of eq. 3, is a time-delay solely because $F$ is smaller than $F_{a}$,
whereas the second term $\tau_{delt}$ is a time-delay due to the barrier
itself, it is the actual tunneling time-delay Kullie (2020). This can be seen
form the factor ${\delta_{z}}/{I_{p}}$, which vanishes for $F=F_{a}$ when the
barrier disappears $\delta_{z}=0$

Indeed, Winful Winful (2003) and Lunardi Lunardi and Manzoni (2019) proposed a
similar unfolded form to the adiabatic tunnel-ionization time-delay given in
eq. .2.

$\displaystyle\tau^{Winf}_{g}$ $\displaystyle=$
$\displaystyle\tau_{si}+\tau_{dwell}$ (5) $\displaystyle\tau^{Lun}_{T}$
$\displaystyle=$ $\displaystyle\tau_{well}+\tau_{barrier}$ (6)

In the quantum tunneling of a wave packet or a flux of particles scattering on
a potential barrier, Winful showed that the group time-delay or the Wigner
time-delay can be written in the form 5, where $\tau_{dwell}$ is the dwell
time which corresponds to our $\tau_{delt}$, and $\tau_{si}$ is according to
Winful a self-interference term, which corresponds to our $\tau_{ion}$.
Importantly, Winful showed that the contribution of the first term is
disentangled from the barrier time-delay Winful (2003). And in their work
Lunardi et. al. used the so-called Slacker-Wigner-Peres quantum-clock (SWP-QC)
with MC-simulation Lunardi and Manzoni (2018, 2019), they found that the
tunneling time in the attoclock is given by the form 6

Figure 2:  (Color online) Illustration of tunnel-ionization of the horizontal
(dashed-dotted, red) and vertical (dashed, blue) channels. The dotted line is
a virtual state. As well as the BSI (green). Compare fig. 1

The author called the first term well-time and, according to the authors, is
the time (-delay) spent before reaching the barrier and suggest that the
second term corresponds to the barrier time (-delay). The three results (ours
confirmed with the NITDSE, Winful’s and Lunardi’s) clearly show that the
separation of eq. .2 presents a unified T-time picture (UTTP) Winful (2003),
allowing us to conclude that the time-delay due the barrier itself is giving
by $\tau_{delt}=\tau_{dwell}=\tau_{barrier}$, which can be determined form eqs
3, .2

$\tau_{{}_{Ad}}-\tau_{dion}=\tau_{delt}=\frac{1}{2}\frac{1}{4Z_{eff}}\frac{\delta_{z}}{F}=\frac{1}{2}\frac{d_{B}}{4Z_{eff}}$
(7)

Eq 7 $\tau_{delt}$ is the barrier time-delay, usually referred to as the time
spent within the barrier $\tau_{barrier}$ or the dwell time $\tau_{dwell}$.
One also finds a linear dependence of $\tau_{delt}$ on the barrier width
$d_{B}$, The barrier time-delay tends to zero in the limit of
$F=F_{a}(\delta_{z}=d_{B}=0)$ because the barrier disappears, as is known from
the seminal work of Hartman Hartman (1962). Obviously, $\tau_{delt}$ cannot be
measured by the experiment directly since the first term is always present,
$\tau_{si},\tau_{well},\tau_{dion}$. However, as seen in eq. 7, it can be
determined taking both field calibrations into account. An illustration of the
tunnel-ionization of adiabatic (horizontal) and nonadiabatic (vertical)
channels is shown in for fig. 2. Fortunately, for the He-atom we can consider
the results of Landsman Landsman  _et al._ (2014) (adiabatic calibration) and
Hofmann Hofmann  _et al._ (2019) (nonadiabatic calibration) since they belong
to the same experiment Eckle  _et al._ (2008a, b). In fig. 3 (left), we show
both experimental results with curves obtained by data fitting. The curves
show a $1/F$-dependence as expected, subtracting them from each other gives
the experimental barrier time $\tau_{barrier,exp}$. In fig. 3 (right), we
compare the barrier time-delay $\tau_{delt}$ of eq. 7 with
$\tau_{barrier,exp}$. We plot $\tau_{delt}$ for two
$Z_{eff}=1.0,1.344=\sqrt{2I_{p}}$, expanding the range to a larger barrier
width (smaller $F$) than specified in the experimental data. As can be seen in
fig. 3, the agreement is very good. It undoubtedly shows that the time spent
in the barrier (or time-delay caused by the barrier upon tunnel-ionization of
the electron) is $\tau_{delt}$, which can be determined from the experimental
data by $\tau_{barrier,exp}$. The agreement becomes better for $Z_{eff}=1$ in
the region of small field strength, because the barrier width
$d_{B}=\delta_{z}/F$ is large and the tunnel-ionized electron escapes far from
the atomic core. Whereas for larger field strength (near the atomic field
strength) the barrier width is small and the agreement is better for
$Z_{eff}=\sqrt{2I_{p}}=1.344$. Finally, all curves tends to zero for
$F\rightarrow F_{a}$, because the barrier disappears, as already mentioned and
should be. Additionally, we plot in fig. 3 a curve ($\tau_{LC}$) of Larmor
clock (LC) time, which is also obtained by fitting the LC data given by
Hofmann Hofmann  _et al._ (2019). The agreement with LC is better for small
$F$ values, which is the limit of thick barrier (more on this in the next
section). This is because Hofmann’s LC time-delay is given for the
nonadiabatic field calibration and our $\tau_{delt}$ becomes closer to
nonadiabatic time-delay $\tau_{dion}$ for small $F$ (thick barrier) (see eq. 8
below), which again explains the disagreement for larger field strength
$F=0.06-0.1$, compare fig. 3. Below the notation $\tau_{delt}$ is used to
refer to the barrier time-delay
$\tau_{delt}(\equiv\tau_{barrier}\equiv\tau_{dwell})$, with the corresponding
adiabatic and nonadiabatic tunnel-ionization time-delay $\tau_{Ad}$,
$\tau_{dion}$, respectively.

##  .3 The weak measurement and the interaction time

In the previous section we found that the LC time $\tau_{LC}$ tends closer to
the barrier time-delay $\tau_{delt},\,\tau_{barrier,exp}$ for small field
strength, compare fig. 3 (right). The LC-time is usually considered in the
context of the so-called weak measurement (WM) approach, which characterize a
system before and after it interacts with a measurement apparatus Aharonov
_et al._ (1988); Rozema  _et al._ (2012). WM does not significantly perturbs
the system, where Landsman Landsman and Keller (2015) showed that the WM value
of the time-delay corresponds to LC-time $\tau_{weak}=\tau_{LC}$ Landsman and
Keller (2015); Steinberg (1995). Thus, for small field strengths in the sense
that $F\ll F_{a}$, the barrier time-delay $\tau_{delt}$ in our model
corresponds to the WM approach, compare fig. 3. Indeed, for $F\ll F_{a}$ the
barrier width $d_{B}=\delta_{z}/F$ becomes too large, which is the limit of
thick barrier. In the limit of thick or opaque barrier, $d_{B}$ approaches the
so-called classical barrier width $d_{C}$, $\lim\limits_{F\ll
F_{a}}d_{B}\approx I_{p}/F=d_{C}$ ($\equiv x_{e,c}$), compare fig. 1. In this
limit the barrier height $\lim\limits_{F\ll F_{a}}{\delta_{z}\to I_{p}}$, and
the barrier time

$\lim\limits_{F\ll
F_{a}}\tau_{delt}=\tau_{a}\frac{F_{a}}{F}\frac{(\delta_{z}\approx
I_{p})}{I_{p}}\approx\tau_{a}\frac{F_{a}}{F}=\tau_{dion}$ (8)

This result is interesting, it shows that the barrier time-delay $\tau_{delt}$
for a thick (or opaque) barrier is approximately equal to tunnel-ionization
time-delay $\tau_{dion}$ of eq. 3, which is the tunnel-ionization time-delay
in the nonadiabatic field calibration Kullie (2024); Hofmann  _et al._ (2019).
This explains the already mentioned agreement between the $\tau_{LC}$ given by
Hofmann for the nonadiabatic field calibration (data-fit curve) and
$\tau_{delt}$ for small field strengths. The agreement is satisfactory
considering that the fitted LC-curve is extended beyond the data range given
in Hofmann  _et al._ (2019).

Figure 3:  (Color online) Left: tunnel-ionization time-delay versus field
strength. Experimental data of adiabatic Landsman  _et al._ (2014) (red) and
nonadiabatic Hofmann  _et al._ (2019) (green) field calibrations with their
data-fit curves. Right: barrier time-delay $\tau_{barrier,exp}$ (orange)
obtained by subtractions of the data-fit curves (shown left), and the barrier
time-delay $\tau_{delt}$ of eq. 7 for $Z_{eff}=1,1.344$ (green, blue). As well
as the LC-curve (magenta) obtained by data-fit of LC-data given in Hofmann
_et al._ (2019).

This leads us to the interaction time with the laser field in the barrier
area, since the LC-time according to Steinberg Steinberg (1995); Spierings and
Steinberg (2021) is related to the interaction time within the barrier, which
corresponds to the time spent in the barrier or the dwell time $\tau_{dwel}$
Ramos  _et al._ (2020); Büttiker (1983), which in turn equals
$\tau_{delt},\tau_{barrier}$ according to the UTTP (and $\tau_{dwel}$ of
Winful in eq. 5).

The agreement shown in Fig. 3 (see eq. 8) suggests that the WM value (LC-time)
and the interaction time within the barrier region in the thick barrier limit
($\tau_{dwell}$), can be determined by $\tau_{delt}$ ($\approx\tau_{dion}$ of
the nonadiabatic field calibration.) We think this is similar to the
measurement presented in Ramos  _et al._ (2020) (Measurement of the time spent
in the barrier with LC). In addition, the back reaction of the measurement of
the system can be found from $\tau_{T,i}$ as the following

$\lim\limits_{F\ll
F_{a}}\tau_{T,i}=\frac{1}{2}\frac{(I_{p}-\delta_{z})}{4Z_{eff}F}=\frac{1}{2}\frac{\varepsilon_{{}_{F}}}{4Z_{eff}F}\approx\frac{2Z_{eff}F}{I_{p}}=\tau_{backr},$

where $\varepsilon_{F}=I_{p}-\delta_{z}$ is small under the WM condition, it
becomes as small as it should be (linear dependence on $F$). The last
approximation in this equation result from the expansion of
$\delta_{z}=I_{p}\sqrt{1-{4Z_{eff}F}/{I_{p}^{2}}}$ for small $F$.

However, the condition of WM is not necessary and we can assume that
$\tau_{T,i}$ always represents the back reaction of the system, which is
generally consistent with interpretation as the time needed to reach the
barrier entrance in the strong-field interaction, see discussion after eq. 2.
Finally, we note that $\tau_{T,d},\tau_{T,i}$ can be interpreted as forward,
backward tunneling, respectively Kullie (2015) and a decomposition of
$\tau_{T,i}$ similar to eq. 2, $\tau_{T,i}=\tau_{dion}-\tau_{delt}$, shows
again that $\tau_{delt}=\tau_{dwell}$, since according to Steinberg Steinberg
(1995) we have $\tau_{dwell}=Re(\tau_{T})=Re(\tau_{R})$, where
$\tau_{T},\tau_{R}$ are the transmission scattering channel and reflection
time respectively Steinberg (1995) and they are forward and backward
(tunneling) scattering channels in our model. In our forward, backward
channels of adiabatic tunneling in eq. 2, which is illustrated in fig. 2
(dashed-dotted, red curve), the condition of a spatially symmetric barrier
noted by Steinberg Steinberg (1995) is reflected by the fact that the
horizontal channel happens along the (same) barrier width
$d_{B}=x_{e,+}-x_{e,-}$ (or the full width of the barrier) for forward and
backward tunneling. Finally we can write
$\tau_{T,{d,i}}=\tau_{dion}\pm\tau_{delt}$ and
$\tau_{dion}=(1/2)\left(\tau_{T,d}+\tau_{T,i}\right),\tau_{delt}=(1/2)\left(\tau_{T,d}-\tau_{T,i}\right)$,
where the symmetrization results in $\tau_{dion}$ and the anti-symmetrization
results in the barrier time-delay or the dwell time. For further details and
discussion, we would like to refer readers to our previous work Kullie (2020).
Forward and backward tunneling is equivalent to the transmission and
reflection of the wave packet in the traditional tunneling or quantum
tunneling studies, which usually use numerical methods Winful (2003); Lunardi
and Manzoni (2019); Dumont and Rivlin (2023); Ma  _et al._ (2024); de Carvalho
and Nussenzveig (2002); Winful (2006), whereas our model offers a simple
tunneling model with expressions composed of basic laser and atomic parameters
that well the measurement result of the attoclock.

### Conclusion

The tunneling time shows a universal behavior that we call UTTP, where the
barrier time-delay can be defined by
$\tau_{delt}\equiv\tau_{barrier}\equiv\tau_{dwell}$. It is shown that
$\tau_{delt}$ agrees with the extracted time-delay $\tau_{barrier,exp}$ from
the difference between adiabatic and nonadiabatic experimental measurement
result of the same experiment. That is the attoclock experiment on He-atom
with the adiabatic and nonadiabatic field calibration of Landsman and Hofmann
Landsman  _et al._ (2014); Hofmann  _et al._ (2019), respectively. Our result
provides conceivable definitions of the tunnel-ionization and barrier time-
delays and the interpretation of the attoclock measurement. We also found that
the barrier time-delay $\tau_{delt}$ corresponds to the LC-time and the
interaction time Büttiker (1983); Steinberg (1995); Ramos  _et al._ (2020);
Spierings and Steinberg (2021). This is particularly evident in the limit of
thick (opaque) barrier, where $\tau_{delt}$ is close to the time-delay in the
nonadiabatic field calibration $\tau_{dion}$, where the back reaction is small
and the weak measurement approach is justified. We assume that there is a
similarity to the measurement presented in Ramos  _et al._ (2020). In the
future, our concern will be devoted to dealing with the superluminal tunnel
effect, e.g. Dumont and Rivlin (2023), one of the most exciting and
controversial phenomena in quantum physics.

### Acknowledgments

I would like to thank Prof. Martin Garcia from the Theoretical Physics of the
Institute of Physics at the University of Kassel for his kind support.

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