# DOUBLY NONNEGATIVE RELAXATION METHOD FOR SOLVING MULTIPLE OBJECTIVE
QUADRATIC PROGRAMMING PROBLEMS

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Multicriterion optimization and Pareto optimality are fundamental tools in
economics. In this paper we propose a new relaxation method for solving
multiple objective quadratic programming problems. Exploiting the technique of
the linear weighted sum method, we reformulate the original multiple objective
quadratic programming problems into a single objective one. Since such single
objective quadratic programming problem is still nonconvex and NP-hard in
general. By using the techniques of lifting and doubly nonnegative relaxation,
respectively, this single objective quadratic programming problem is
transformed to a computable convex doubly nonnegative programming problem. The
optimal solutions of this computable convex problem are (weakly) Pareto
optimal solutions of the original problem under some mild conditions.
Moreover, the proposed method is tested with two examples and a practical
portfolio selection problem. The test problems are solved by CVX package which
is a solver for convex optimization. The numerical results show that the
proposed method is effective and promising.

###### Key words and phrases:

Multiple objective programming, quadratic programming, linear weighted sum
method, copositive programming, completely positive programming

###### 1991 Mathematics Subject Classification:

Primary: 90C29, 90C26; Secondary: 49M20

The research is supported by National Natural Science Foundation of China (No.
11071158) and the Key Disciplines of Shanghai Municipality (No. S30104).

yanqin bai

Department of Mathematics, Shanghai University

Shanghai 200444, China

chuanhao guo

Department of Mathematics, Shanghai University

Shanghai 200444, China

(Communicated by the associate editor name)

##  1\. Introduction

We consider multiple objective nonconvex quadratic programming problems as
follows

$\rm{(MOQP)}\begin{array}[]{llll}&\min&F(x)=(F_{1}(x),F_{2}(x),\ldots,F_{p}(x))^{T}&\\\
&{\rm s.t.}&Ax=b,\\\ &&x\geq 0,&\\\ \end{array}$

where $F_{i}(x)=x^{T}Q_{i}x+2c_{i}^{T}x$ and $x\in R^{n}$ is the decision
variable. $Q_{i}\in R^{n\times n},\ c_{i}\in R^{n},\ i=1,\ldots,p$,
$A=(a_{1},a_{2},\ldots,a_{m})^{T}\in R^{m\times n}$ and
$b=(b_{1},b_{2},\ldots,b_{m})\in R^{m}$ are given data. Without loss of
generality, $Q_{i}$ is symmetric and not positive semidefinite by assumption.

Multi-objective programming (MOP) also known as multi-criteria optimization,
is the process of simultaneously optimizing two or more conflicting objectives
subject to certain constraints. (MOP) problems are found in many fields, such
as facility location and optimal detector design [5], image processing [8].
Problem (MOQP) is a subclass of (MOP) problem and arises in portfolio
selection [16], reservoir optimal operation [18] and so on. Problem (MOQP)
also can be viewed as an extension of multiple objective quadratic-linear
programming (MOQLP) problem for which the objectives are a quadratic and
several linear functions and the constraints are linear functions which were
studied in [13, 16].

Problem (MOP) does not have a single solution that simultaneously minimizes
each objective function. A tentative solution is called Pareto optimal if it
impossible to make one objective function better off without necessarily
making the others worse off. And problem (MOP) may have many Pareto optimal
solutions. For solving problem (MOP), the linear weighted sum method is one of
the most widely used methods. The main idea is to choose the weighting
coefficients corresponding to objective functions. Then, problem (MOP) can be
transformed to a single objective one, and the Pareto optimal solutions for
problem (MOP) could be found by solving this single objective problem with the
appropriate weights. Ammar [1, 2] investigates problem (MOQP) with fuzzy
random coefficient matrices. Under the assumption that the coefficient
matrices in objectives are positive semidefinite, some results are discussed
to deduce Pareto optimal solutions for fuzzy problem (MOQP). However, many
practical problems of this class of problems are nonconvex in general. So,
these two methods have certain limitations in practical applications.

Burer [6] proves that a large class of NP-hard nonconvex quadratic program
with a mix of binary and continuous variables can be modeled as so called
completely positive programs (CPP), i.e., the minimization of a linear
function over the convex cone of completely positive matrices subject to
linear constraints (For more details and developments of this technique, one
may refer to [4, 6, 7, 14]). In order to solve such convex programs
efficiently, a computable relaxed problem is obtained by approximation the
completely positive matrices with doubly nonnegative matrices, resulting in a
doubly nonnegative programming [7], which can be efficiently solved by some
popular packages.

Motivated by the ideas of [6, 7], we propose a new relaxation method for
solving problem (MOQP) by combining with the linear weighted sum method. First
of all, in virtue of the linear weighted sum method, we first transform
problem (MOQP) into a single objective quadratic programming (SOQP) problem
over a linearly constrained subset of the cone of nonnegative orthant. Since
problem (SOQP) is a nonconvex in general, which is equivalently reformulated
as a completely positive programming problem, which is NP-hard. Furthermore, a
computable relaxed convex problem for this completely positive programming
problem is derived by using doubly nonnegative relaxation technique, and
resulting in a doubly nonnegative programming (DNNP) problem. Based on the
characteristics of optimal solutions of problem (DNNP), a sufficient condition
for (weakly) Pareto optimal solutions for problem (MOQP) is proposed.
Moreover, the proposed method is tested with two examples and a practical
portfolio selection problem. The test problems are solved by CVX package,
which is a solver for convex optimization. The numerical results show that the
proposed method is effective and promising.

The paper is organized as follows. In Section 2, we recall some basic
definitions and preliminaries for (weakly) Pareto optimal solution and the
linear weighted sum method, respectively. In Section 3, problem (MOQP) is
transformed into problem (SOQP) by using the linear weighted sum method. And
some optimality conditions for problem (MOQP) are established. In order to
solve problem (SOQP) effectively, problem (SOQP) is equivalently reformulated
as a convex problem (CP) which is further relaxed to a computable (DNNP)
problem in Section 4. In Section 5, numerical results are given to show the
performance of the proposed method. Some conclusions and remarks are given in
Section 6.

###  1.1. Notation and terminology

Let $\Omega=\\{x\in R^{n}:Ax=b,x\geq 0\\}$ be the feasible set of problem
(MOQP). Let $R^{n}_{+}$ (or $R^{n}_{++}$) denotes the cone of nonnegative (or
positive) vectors with dimension $n$, $S_{n}$ the cone of all $n\times n$
symmetric matrices, $S_{n}^{+}$ the cone of all $n\times n$ symmetric positive
semidefinite matrices and $(S_{n})^{+}$ the cone of all $n\times n$ symmetric
matrices with nonnegative elements. $C_{n}^{*}$ is the cone of all $n\times n$
completely positive matrices, i.e.,

$C_{n}^{*}=\\{X\in R^{{n}\times{n}}:\ X=\sum\limits_{k\in
K}z^{k}(z^{k})^{T}\\}\cup\\{0\\},$

where $\\{z^{k}\\}_{k\in K}\subset R^{n}_{+}\backslash\\{0\\}$. For two
vectors $x,\ y\in R^{n}$, $x\circ y$ is a vector in $R^{n}$ with $x_{i}y_{i}$
is its $i$-th component. For a matrix $M$, ${\rm{diag}(M)}$ is a column vector
whose elements are the diagonal elements of $M$. Given two conformal matrices
$A$ and $B$, $A\bullet B={\rm trace}(A^{T}B)$. For a given optimization
problem $(\ast)$, its optimal objective value is denoted by $\rm{Opt(\ast)}$.

##  2\. Preliminaries

###  2.1. Pareto optimal solutions

In multi-objective optimization with conflicting objectives, there is no
unique optimal solution. A simple optimal solution may exist here only when
the objectives are non-conflicting. For conflicting objectives one may at best
obtain what is called Pareto optimal solutions. For the sake of completeness,
we restate the definitions of some types of Pareto optimal solutions and ideal
point from [12].

######  Definition 2.1.

A solution $x^{*}$ is said to be Pareto optimal solution of problem (MOQP) if
and only if there does not exist another feasible solution $x\in\Omega$ such
that $F_{i}(x)\leq F_{i}(x^{*})$ for all $i=1,\ldots,p$ and
$F_{i}(x)<F_{i}(x^{*})$ for at least one index $i$.

All Pareto optimal points lie on the boundary of the feasible region $\Omega$.
Often, algorithms provide solutions that may not be Pareto optimal but may
satisfy other criteria, making them significant for practical applications.
For instance, weakly Pareto optimal is defined as follows.

######  Definition 2.2.

A solution $x^{*}$ is said to be weakly Pareto optimal solution of problem
(MOQP) if and only if there does not exist another feasible solution
$x\in\Omega$ such that $F_{i}(x)<F_{i}(x^{*})$ for all $i=1,\ldots,p$.

######  Remark 1.

A solution is weakly Pareto optimal if there is no other point that improves
all of the objective functions simultaneously. In contrast, a point is Pareto
optimal if there is no other point that improves at least one objective
function without detriment to another function. It is obvious that each Pareto
optimal point is weakly Pareto optimal, but weakly Pareto optimal point is not
Pareto optimal.

In order to illustrate (weakly) Pareto optimal solution intuitively, two
examples are given as follows.

######  Example 1.

Let $\Omega=[0,3]$, and

$F_{1}(x)=\frac{x^{2}}{2}+1,\
F_{2}(x)=\left\\{\begin{array}[]{llll}-x+2,&x\in[0,1],\\\ 1,&x\in(1,2],\\\
x-1,&x\in(2,3].\end{array}\right.$

Figure 1. Pareto optimal solution set [0, 1] and weakly Pareto optimal
solution set [0, 2]

The design space for this problem is shown in Figure 1. According to the above
two definitions about (weakly) Pareto optimal solution, we can easily get
Pareto optimal solution set for this problem is $[0,1]$, and weakly Pareto
optimal solution set is $[0,2]$. Note that each Pareto optimal solution is
weakly Pareto optimal solution for this problem, since $[0,1]\subseteq[0,2]$,
this also shows the conclusion holds in Remark 1.

######  Example 2.

Let $\Omega=[0,4]$, and

$F_{1}(x)=(x-1)^{2}+2,\ F_{2}(x)=(x-2)^{2}+1,\ F_{3}(x)=\|x\|.$

Figure 2. Pareto optimal solution set [0, 2] and weakly Pareto optimal
solution set [0, 2]

For this problem, the graphs for each objective function is plotted in Figure
2. By Definitions 2.1 and 2.2, it holds that Pareto optimal solution and
weakly Pareto optimal solution for this problem are the same, they all equal
to $[0,2]$.

###  2.2. The linear weighted sum method

One useful way of getting the efficiency of problem (MOQP) is to build a
utility function [17] according to the decision makers provided preference
information, such that each solution gained by this method is Pareto optimal
solution of problem (MOQP).

The linear weighted function

$u(F(\cdot)):=\sum\limits_{i=1}^{p}\lambda_{i}F_{i}(\cdot)$ (1)

is one of the most widely used utility function, where weight $\lambda_{i}$
corresponding to objective functions $F_{i}(\cdot),\ i=1,\ldots,p$ satisfy the
following conditions

$\sum\limits_{i=1}^{p}\lambda_{i}=1,\ \lambda_{i}\geq 0,\ i=1,\ldots,p,$ (2)

which is provided by the decision makers, and weights $\lambda_{i}$ imply that
the relative importance for $F_{i}(\cdot)$ in the heart of the decision
makers.

##  3\. Optimality conditions

In this section, we derive a single objective quadratic programming (SOQP)
problem corresponding to problem (MOQP) by the linear weighted sum method.
And, some optimality conditions for problem (MOQP) are proposed based on the
optimal solutions for problem (SOQP).

From (1) and (2), problem (MOQP) can be convert to a single objective
quadratic programming problem as follows

$\rm{(SOQP)}\begin{array}[]{llll}&\min&\sum\limits_{i=1}^{p}\lambda_{i}(x^{T}Q_{i}x+2c_{i}^{T}x)&\\\
&{\rm s.t.}&Ax=b,\\\ &&x\geq 0.&\\\ \end{array}$

Similar to Corollary 3.1 in [17], we can get the following theorem, its proof
can be found in [17].

######  Theorem 3.1.

Let $x^{*}$ be an optimal solution for problem (SOQP), it follows that $x^{*}$
is a Pareto optimal solution (weakly Pareto optimal solution) for problem
(MOQP) if weight $\lambda>0$ ($\lambda\geq 0$, $\lambda\neq 0$).

According to the above Theorem 3.1, varying weight $\lambda$ consistently and
continuously can result in a subset of Pareto optimal (weakly Pareto optimal)
set for problem (MOQP). The following theorem presents a sufficient optimality
condition for weakly Pareto solution of problem (MOQP).

Firstly, we quote the following definition which will be used in the sequel.

######  Definition 3.2.

Let $\bar{x}\in\Omega$ and $Y=F(\Omega)$, if there exists a convex set $H$
such that

$Y\subseteq H,\ {\rm{and}}\ H\cap(F(\bar{x})-R^{p}_{++})=\emptyset,$

then we say that $Y$ satisfies the convex inclusion condition at $F(\bar{x})$.

Based on the above convex inclusion condition, we have the following theorem.
The proof is omitted here for the reason that it is similar to the one of
Theorem 3.6 in [17].

######  Theorem 3.3.

Let $x^{*}$ be a weakly Pareto optimal solution for problem (MOQP), and $Y$
satisfies the convex inclusion condition at $F(x^{*})$. Then there exists a
weight $\bar{\lambda}\geq 0,\ \bar{\lambda}\neq 0$ such that $x^{*}$ is an
optimal solution for problem (SOQP) with $\bar{\lambda}$.

In particular, if problem (MOQP) is convex, Theorem 3.3 still holds without
the convex inclusion condition.

##  4\. Reformulation

Note that problem (SOQP) is a nonconvex quadratic programming problem in
general, and thus it is NP-hard. If problem (SOQP) is convex with appropriate
coefficient $\lambda$, we may use some popular convex packages to solve it
directly. In the following section, we will establish the computable convex
reformulation for problem (SOQP) when it is nonconvex, and the details are as
follows.

###  4.1. Completely positive reformulation

Motivated by the ideas in [6], problem (SOQP) can be reformulated as a
completely positive programming problem. First, the definition of completely
positive [3] is given as follows.

######  Definition 4.1.

A symmetric matrix $B$ of order $n$ is called completely positive if one can
find an integer $m$ and a matrix $V$ of size $n\times m$ with nonnegative
entries such that $B=VV^{T}$, where the smallest possible number $m$ is called
the CP-rank of $B$.

Based on above definition for completely positive, by using the techniques in
[6], problem (SOQP) can be reformulated as the following completely positive
programming problem

$\rm{(CP)}\begin{array}[]{llll}&\min&\sum\limits_{i=1}^{p}\lambda_{i}(Q_{i}\bullet
X+2c_{i}^{T}x)\\\ &{\rm s.t.}&Ax=b,\\\ &&{\rm{diag}}(AXA^{T})=b\circ b,\\\
&&\left(\begin{array}[]{lll}1&x^{T}\\\ x&X\end{array}\right)\in
C_{1+n}^{*},\end{array}$

which is a convex programming problem. Similar to Theorem 2.6 in [6], the
following theorem holds immediately, for more details can be seen in [6].

######  Theorem 4.2.

$\it{Opt}(SOQP)=\it{Opt}(CP)$, and if $(x^{*},X^{*})$ is an optimal solution
for problem (CP), then $x^{*}$ is in the convex hull of optimal solutions of
problem (SOQP).

According to Theorem 4.2, problem (SOQP) is equivalent to problem (CP).
However, problem (CP) is NP-hard, since there is a cone $C^{*}_{1+n}$
constraint, and check whether or not a given matrix belong to $C^{*}_{1+n}$ is
shown to be NP-hard [10], one must relax it in practice. Relaxing problem (CP)
in a natural way yields a doubly nonnegative programming (DNNP) problem.

###  4.2. Doubly nonnegative relaxation

As mentioned above, in order to establish the doubly nonnegative relaxation
for problem (CP), the definition of doubly nonnegative is given as follows.

######  Definition 4.3.

If matrix $Z\in S_{n}$ is not only nonnegative but also positive semidefinite,
then $Z$ is called doubly nonnegative.

Note that if $Z\in C^{*}_{n}$, it necessarily holds that $Z$ is doubly
nonnegative from the above Definitions 4.1 and 4.3. Moreover, the convex cone
$S_{n}^{+}$ is self-dual, and so is the convex cone $(S_{n})^{+}$. Hence,
Diananda’s decomposition theorem [9] can be reformulated as follows.

######  Theorem 4.4.

For all $n$, we have $C_{n}^{*}\subseteq S_{n}^{+}\cap(S_{n})^{+}$. The
relationship $"\supseteq"$ for two sets holds if and only if $n\leq 4$.

Regardless of the dimension $n$, one always has the inclusion
$C_{n}^{*}\subseteq S_{n}^{+}\cap(S_{n})^{+}$. Of course, in dimension $n\geq
5$ there are matrices which are doubly nonnegative but not completely
positive. The counterexample

$M=\left[\begin{array}[]{lllllll}1&1&0&0&1\\\ 1&2&1&0&0\\\ 0&1&2&1&0\\\
0&0&1&2&1\\\ 1&0&0&1&6\end{array}\right]$

proposed by Diananda [9] to illustrates this point.

Replacing $C^{*}_{1+n}$ by $S_{1+n}^{+}\cap(S_{1+n})^{+}$ according to Theorem
4.4, problem (CP) is relaxed to the following doubly nonnegative programming
problem

$\rm{(DNNP)}\begin{array}[]{llll}&\min&\sum\limits_{i=1}^{p}\lambda_{i}(Q_{i}\bullet
X+2c_{i}^{T}x)\\\ &{\rm s.t.}&Ax=b,\\\ &&{\rm{diag}}(AXA^{T})=b\circ b,\\\
&&\left(\begin{array}[]{lll}1&x^{T}\\\ x&X\end{array}\right)\in
S_{1+n}^{+}\cap(S_{1+n})^{+},\end{array}$

which is not only a convex problem but also can be solved in polynomial time
to any fixed precision from the theory of interior-point methods.

Up to now, problem (MOQP) is reformulated as above problem (DNNP), which can
be solved by some popular package CVX. It is obviously that problem (DNNP) is
a relaxation form for problem (MOQP).

In the last of this section, we will investigate the relationship between
optimal solutions for problems (MOQP) and (DNNP), i.e., a sufficient condition
for (weakly) Pareto optimal solutions of problem (MOQP) based on the
characteristics of optimal solutions for problem (DNNP) is established in the
following part.

######  Theorem 4.5.

Let $(x^{*},X^{*})$ be an optimal solution for problem (DNNP). If the
relationship $X^{*}=x^{*}(x^{*})^{T}$ holds, then
${\it{Opt(DNNP)}}={\it{Opt(CP)}}.$ Moreover, $(x^{*},X^{*})$ is an optimal
solution for problem (CP).

###### Proof.

On one hand, from Theorem 4.4, it is obviously holds that

${\rm{Opt}(CP)}\geq{\rm{Opt}(DNNP)}.$ (3)

On the other hand, from $X^{*}=x^{*}(x^{*})^{T}$, and constraints of problem
(CP), we have $(x^{*},X^{*})$ is also a feasible solution for problem (CP).
Since problems (DNNP) and (CP) have the same objective function, it follows
that

${\rm{Opt(CP)}}\leq\sum\limits_{i=1}^{p}\lambda_{i}(Q_{i}\bullet
X^{*}+2c_{i}^{T}x^{*})={\rm{Opt(DNNP)}}.$ (4)

Thus, combining (3) and (4), we have

${\rm{Opt(CP)}}={\rm{Opt(DNNP)}}.$

Again from problems (CP) and (DNNP) have the same objective function, it holds
that $(x^{*},X^{*})$ is an optimal solution for problem (CP). ∎

######  Remark 2.

It holds that ${\it{Opt(SOQP)}}={\it{Opt(CP)}}$ from Theorem 4.2. Let
$(x^{*},X^{*})$ be an optimal solution for problem (DNNP), if
$X^{*}=x^{*}(x^{*})^{T}$, by Theorem 4.5, we have
${\it{Opt(CP)}}={\it{Opt(DNNP)}}.$ Thus, we get
${\it{Opt(SOQP)}}={\it{Opt(DNNP)}}$ under the condition
$X^{*}=x^{*}(x^{*})^{T}$. Furthermore, since problems (SOQP) and (DNNP) have
the same objective function, again from Theorem 4.2, we can conclude that
$x^{*}$ is an optimal solution for problem (SOQP). From Theorem 3.1, we
further know that $x^{*}$ is a Pareto optimal solution (or weakly Pareto
optimal solution) for problem (MOQP) if $\lambda>0$ (or $\lambda\geq 0$).

##  5\. Numerical experiments

In this section, in order to show the effectiveness of our proposed method,
some examples are tested and corresponding numerical results are reported. To
solve test problems, we use CVX [11], a package for specifying and solving
convex programs. The software is implemented using MATLAB R2011b on Windows 7
platform, and on a PC with Intel(R) Core(TM) i3-2310M CPU 2.10 GHz.

In the following numerical experiments, three examples are solved by using the
proposed method, respectively. The first example is a given two-dimension
problem, which has four nonconvex objective functions. The second example is a
five-dimension nonconvex problem with five objective functions. Note that its
coefficients are generated by MATLAB function randn($\cdot$). The last example
is a practical portfolio selection problem, which is taken from [16]. The
weighted coefficient $\lambda$ is generated by the following procedure

lambda=zeros(p,1);

while lambda(p)==0

lambda(1:p-1)=rand(p-1,1); s=sum(lambda);

if s<1 lambda(p)=1-s; end

end
where $p$ is the number of objective functions.

######  Example 3.

First of all, a two-dimension problem with four objective functions is tested.
The corresponding coefficients $Q_{i},\ c_{i},\ i=1,\ldots,4$ and $A,\ b$ are
given in Table 1.

Table 1. Coefficients for Example 3 $Q_{1}=\left[\begin{array}[]{ll}1&3\\\ 3&8\end{array}\right]$ | $Q_{2}=\left[\begin{array}[]{lllll}2&0\\\ 0&-2\end{array}\right]$ | $Q_{3}=\left[\begin{array}[]{lllll}0&2\\\ 2&0\end{array}\right]$ | $Q_{4}=\left[\begin{array}[]{llllll}-1&0\\\ 0&0\end{array}\right]$
---|---|---|---
$c_{1}=[2.5\ \ -0.5]^{T}$ | $c_{2}=[-1\ \ -1.5]^{T}$ | $c_{3}=[1\ \ 0.5]^{T}$ | $c_{4}=[1\ \ 1.5]^{T}$
$A=[1\ \ 1]$ | $b=1$ |  |

First, by relative simple computation, we obtain optimal solutions for each
objective function which is minimized independently. The corresponding optimal
numerical results of each objective function are given in Table 2.

Table 2. Optimal numerical results for each objective function of Example 3 | $F_{1}(x^{*})$ |  |  |  |  |  $F_{2}(x^{*}$) |  |  |  |  |  $F_{3}(x^{*}$) |  |  |  |  |  $F_{4}(x^{*}$)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
FV | $5.6667$ |  |  |  |  | $-5$ |  |  |  |  | $1$ |  |  |  |  | $1$
$x^{*}$ | $(0.6667\ 0.3333)^{T}$ |  |  |  |  | $(0\ 1)^{T}$ |  |  |  |  | $(0\ 1)^{T}$ |  |  |  |  | $(1\ 0)^{T}$

In Table 2, the labels FV and $x^{*}$ denotes optimal values and optimal
solutions corresponding to each objective function, respectively. The results
in Table 2 show that the objective functions $F_{2}(x)$ and $F_{3}(x)$ have
the same optimal solution, which is different from the other two functions
optimal solutions. These imply that we can not find a single solution that
simultaneously optimizes each objective function. Moreover, note that these
four optimal solutions corresponding to each objective function are all weakly
Pareto optimal solutions for Example 3.

Table 3. Optimal numerical results for Example 3 in (DNNP) $\lambda$ | FV | $x^{*}$ | $X^{*}$
---|---|---|---
$(0.5472\ 0.1386\ 0.1493\ 0.1649)$ | $\left[\begin{array}[]{ll}5.8345\\\ -2.8494\\\ 2.4106\\\ 2.3849\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4301\\\ 0.5699\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1850&0.2451\\\ 0.2451&0.3248\end{array}\right]$
$(0.3500\ 0.1966\ 0.2511\ 0.2023)$ | $\left[\begin{array}[]{ll}7.0000\\\ -5.0000\\\ 1.0000\\\ 3.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000&0.0000\\\ 0.0000&1.0000\end{array}\right]$
$(0.0759\ 0.0540\ 0.5308\ 0.3394)$ | $\left[\begin{array}[]{ll}6.0000\\\ 0.0000\\\ 2.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000\\\ 0.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000&0.0000\\\ 0.0000&0.0000\end{array}\right]$
$(0.2417\ 0.4039\ 0.0965\ 0.2579)$ | $\left[\begin{array}[]{ll}5.6670\\\ -1.6105\\\ 2.5513\\\ 1.8625\end{array}\right]$ | $\left[\begin{array}[]{ll}0.6779\\\ 0.3221\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4596&0.2183\\\ 0.2183&0.1037\end{array}\right]$
$(0.5752\ 0.0598\ 0.2348\ 0.1302)$ | $\left[\begin{array}[]{ll}5.8570\\\ -2.9261\\\ 2.3857\\\ 2.4132\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4148\\\ 0.5852\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1720&0.2427\\\ 0.2427&0.3425\end{array}\right]$
$(0.0430\ 0.1690\ 0.6491\ 0.1389)$ | $\left[\begin{array}[]{ll}7.0000\\\ -5.0000\\\ 1.0000\\\ 3.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000&0.0000\\\ 0.0000&1.0000\end{array}\right]$
$(0.2259\ 0.1707\ 0.2277\ 0.3757)$ | $\left[\begin{array}[]{ll}6.0000\\\ 0.0000\\\ 2.0000\\\ 1.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000\\\ 0.0000\end{array}\right]$ | $\left[\begin{array}[]{ll}1.0000&0.0000\\\ 0.0000&0.0000\end{array}\right]$
$(0.2920\ 0.4317\ 0.0155\ 0.2609)$ | $\left[\begin{array}[]{ll}5.7480\\\ -2.4901\\\ 2.5020\\\ 2.2460\end{array}\right]$ | $\left[\begin{array}[]{ll}0.5020\\\ 0.4980\end{array}\right]$ | $\left[\begin{array}[]{ll}0.2520&0.2500\\\ 0.2500&0.2480\end{array}\right]$
$(0.2607\ 0.5944\ 0.0225\ 0.1224)$ | $\left[\begin{array}[]{ll}5.7480\\\ -2.4901\\\ 2.5020\\\ 2.2460\end{array}\right]$ | $\left[\begin{array}[]{ll}0.5020\\\ 0.4980\end{array}\right]$ | $\left[\begin{array}[]{ll}0.2520&0.2500\\\ 0.2500&0.2480\end{array}\right]$

It is very easy to verify that the given four objective functions are all
nonconvex by using MATLAB function eig. Furthermore, we obtain problem (SOQP)
is nonconvex with corresponding coefficients $\lambda$ which proposed in Table
3 by using eig. Hence, we use our method to solve this problem. The
corresponding optimal numerical results are reported in Table 3. In Table 3,
Example 3 is transformed into problem (DNNP), and then is solved with nine
different weighted coefficient $\lambda$. The results of $x^{*}$ and $X^{*}$
in Table 3 show that $X^{*}=x^{*}(x^{*})^{T}$ holds for nine different
weighted coefficients. Thus, from Theorem 4.5 and Remark 2, it holds that each
optimal solution $x^{*}$ in Table 3 also is Pareto optimal solution for
Example 3. Moreover, we obtain weakly Pareto optimal solutions for Example 3
when the weighted coefficient $\lambda$ is chosen appropriately. For instance,
if $\lambda$ is chosen as $(0.3500,0.1966,0.2511,0.2023)$ or
$(0.0430,0.1690,0.6491,0.1389)$, then weakly Pareto optimal solution
$(0.0000,1.0000)^{T}$ is obtained by using our method.

######  Example 4.

In this test problem, we set $n=5$, $m=4$ and $p=5$. The corresponding
coefficients $Q_{i},\ c_{i},\ i=1,\ldots,5$ and $A,\ b$ are generated by the
functions tril(randn(n,n),-1)+triu(randn(n,n)’,0), randn(n,1), randn(n,m) and
randn(m,1), respectively, and the details can be seen in Table 4.

Table 4. Coefficients for Example 4 $Q_{1}=\left[\begin{array}[]{lllllll}0.4056&-1.4193&-0.7294&1.1473&0.5979\\\ -1.4193&-2.2033&-0.5712&0.2140&0.9424\\\ -0.7294&-0.5712&0.3062&-1.1723&-0.9610\\\ 1.1473&0.2140&-1.1723&-0.9000&-0.2857\\\ 0.5979&0.9424&-0.9610&-0.2857&0.6103\end{array}\right]$ | $c_{1}=\left[\begin{array}[]{ll}0.7907\\\ 0.2877\\\ 0.0032\\\ 0.3656\\\ 3.5267\end{array}\right]$
---|---
$Q_{2}=\left[\begin{array}[]{lllll}0.0591&-1.4669&-1.6258&-1.9648&2.6052\\\ -1.4669&0.2570&-0.9742&-1.1464&0.5476\\\ -1.6258&-0.9742&-0.4494&-0.0843&-1.9920\\\ -1.9648&-1.1464&-0.0843&-0.3909&0.4092\\\ 2.6052&0.5476&-1.9920&0.4092&1.3018\end{array}\right]$ | $c_{2}=\left[\begin{array}[]{ll}-0.1124\\\ -1.5566\\\ 1.9151\\\ 0.6098\\\ -0.6479\end{array}\right]$
$Q_{3}=\left[\begin{array}[]{lllll}-0.5936&0.4364&-0.5044&0.1021&1.1963\\\ 0.4364&-1.0368&-0.8571&-0.1699&-0.1917\\\ -0.5044&-0.8571&1.2665&-0.2512&-0.2046\\\ 0.1021&-0.1699&-0.2512&-0.3862&0.5256\\\ 1.1963&-0.1917&-0.2046&0.5256&0.8175\end{array}\right]$ | $c_{3}=\left[\begin{array}[]{ll}2.6173\\\ 0.5510\\\ 0.2942\\\ -0.7778\\\ -1.0649\end{array}\right]$
$Q_{4}=\left[\begin{array}[]{llllll}0.4902&0.7653&0.7783&-1.4803&0.5404\\\ 0.7653&-0.7603&-0.6936&1.2815&-0.8097\\\ 0.7783&-0.6936&2.0108&0.0256&0.3083\\\ -1.4803&1.2815&0.0256&0.5301&-0.9521\\\ 0.5404&-0.8097&0.3083&-0.9521&-0.4506\end{array}\right]$ | $c_{4}=\left[\begin{array}[]{ll}-1.7684\\\ -0.4229\\\ -1.0531\\\ 0.6478\\\ -0.3176\end{array}\right]$
$Q_{5}=\left[\begin{array}[]{lllllll}0.1092&-0.2506&-0.1899&-1.0329&-0.3233\\\ -0.2506&1.7447&-1.1605&2.3774&1.5261\\\ -0.1899&-1.1605&-0.6987&0.8328&-0.6946\\\ -1.0329&2.3774&0.8328&0.8967&0.5047\\\ -0.3233&1.5261&-0.6946&0.5047&1.1867\end{array}\right]$ | $c_{5}=\left[\begin{array}[]{ll}1.7690\\\ 1.5106\\\ 0.1640\\\ -0.2828\\\ 1.1522\end{array}\right]$
$A=\left[\begin{array}[]{llllll}-1.1465&-0.6718&0.5530&0.2695&1.0393\\\ 0.6737&0.5756&-0.4234&-2.5644&0.9109\\\ -0.6691&-0.7781&0.3616&0.4659&-0.2397\\\ -0.4003&-1.0636&-0.3519&1.8536&0.1810\end{array}\right]$ | $b=\left[\begin{array}[]{ll}0.2442\\\ 0.0964\\\ -0.8305\\\ -0.3523\end{array}\right]$

In order to verify that whether the given objective functions in Table 4 are
nonconvex functions or not, the corresponding eigenvalues for $Q_{i},\
i=1,\ldots,5$ are given in Table 5. The results in Table 5 show that five
objective functions are all nonconvex. Thus, we will use the proposed method
to solve this problem. The results are given in Tables 6 and 7.

Table 5. Eigenvalues for $Q_{i},\ i=1,\ldots,5$ in Example 4 Quadratic matrices | Eigenvalues
---|---
$Q_{1}$ | $(-3.3892,-1.8503,-0.0780,1.1180,2.4184)$
$Q_{2}$ | $(-4.2248,-1.8386,0.7234,1.3339,4.7837)$
$Q_{3}$ | $(-1.6777,-1.1336,-0.3917,1.1973,2.0731)$
$Q_{4}$ | $(-2.5335,-1.0738,0.6848,1.6750,3.0677)$
$Q_{5}$ | $(-2.3512,-0.6382,0.1606,1.3837,4.6838)$
Table 6. Optimal numerical results for each objective function of Example 4 |  |  |  |  | FV |  |  |  |  | $x^{*}$
---|---|---|---|---|---|---|---|---|---|---
$F_{1}(x^{*})$ |  |  |  |  | $2.1900$ |  |  |  |  | $(0,\ 1.2165,\ 0.2253,\ 0.4747,\ 0.7783)^{T}$
$F_{2}(x^{*})$ |  |  |  |  | $-9.2542$ |  |  |  |  | $(2.7216,\ 0,\ 2.7773,\ 0.7727,\ 1.5592)^{T}$
$F_{3}(x^{*})$ |  |  |  |  | $-1.3755$ |  |  |  |  | $(0,\ 1.2165,\ 0.2253,\ 0.4747,\ 0.7783)^{T}$
$F_{4}(x^{*})$ |  |  |  |  | $-2.2276$ |  |  |  |  | $(0.4915,\ 0.9969,\ 0.6861,\ 0.5285,\ 0.9193)^{T}$
$F_{5}(x^{*})$ |  |  |  |  | $0.6759$ |  |  |  |  | $(2.7216,\ 0,,\ 2.7773,\ 0.7727,\ 1.5592)^{T}$

Table 6 shows that the optimal numerical results for each objective function
of Example 4. The results in Table 6 show that we can not find a single
solution that simultaneously optimizes these five objective functions. Note
that these five optimal solutions are also weakly Pareto optimal solutions for
Example 4.

Table 7. Optimal numerical results for Example 4 in (DNNP) $\lambda$ | FV | $x^{*}$ | $X^{*}$
---|---|---|---
$\left[\begin{array}[]{ll}0.3317\\\ 0.1522\\\ 0.3480\\\ 0.1217\\\ 0.0464\end{array}\right]$ | $\left[\begin{array}[]{llllll}2.1900\\\ -1.7656\\\ -1.3755\\\ -1.7914\\\ 7.0248\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.2165\\\ 0.2253\\\ 0.4747\\\ 0.7783\end{array}\right]$ | $\left[\begin{array}[]{lllll}0.0000&0.0000&0.0000&0.0000&0.0000\\\ 0.0000&1.4799&0.2741&0.5775&0.9469\\\ 0.0000&0.2741&0.0508&0.1070&0.1753\\\ 0.0000&0.5775&0.1070&0.2254&0.3695\\\ 0.0000&0.9469&0.1753&0.3695&0.6058\end{array}\right]$
$\left[\begin{array}[]{ll}0.0712\\\ 0.1820\\\ 0.0930\\\ 0.4635\\\ 0.1903\end{array}\right]$ | $\left[\begin{array}[]{lll}2.2944\\\ -2.3834\\\ 0.1927\\\ -2.2196\\\ 6.2356\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4250\\\ 1.0266\\\ 0.6238\\\ 0.5213\\\ 0.9003\end{array}\right]$ | $\left[\begin{array}[]{lllllll}0.1806&0.4363&0.2651&0.2215&0.3826\\\ 0.4363&1.0538&0.6403&0.5351&0.9242\\\ 0.2651&0.6403&0.3891&0.3252&0.5616\\\ 0.2215&0.5351&0.3252&0.2717&0.4693\\\ 0.3826&0.9242&0.5616&0.4693&0.8105\end{array}\right]$
$\left[\begin{array}[]{ll}0.0290\\\ 0.6371\\\ 0.0595\\\ 0.1692\\\ 0.1052\end{array}\right]$ | $\left[\begin{array}[]{ll}2.3045\\\ -2.4426\\\ 0.3234\\\ -2.2257\\\ 6.1681\end{array}\right]$ | $\left[\begin{array}[]{ll}0.4595\\\ 1.0111\\\ 0.6562\\\ 0.5251\\\ 0.9102\end{array}\right]$ | $\left[\begin{array}[]{llllll}0.2111&0.4646&0.3015&0.2413&0.4182\\\ 0.4646&1.0224&0.6635&0.5309&0.9203\\\ 0.3015&0.6635&0.4305&0.3445&0.5972\\\ 0.2413&0.5309&0.3445&0.2757&0.4779\\\ 0.4182&0.9203&0.5972&0.4779&0.8284\end{array}\right]$
$\left[\begin{array}[]{ll}0.0472\\\ 0.0559\\\ 0.0715\\\ 0.0157\\\ 0.8098\end{array}\right]$ | $\left[\begin{array}[]{ll}3.5136\\\ -9.2542\\\ 9.9301\\\ 6.7545\\\ 0.6759\end{array}\right]$ | $\left[\begin{array}[]{ll}2.7216\\\ 0.0000\\\ 2.7773\\\ 0.7727\\\ 1.5592\end{array}\right]$ | $\left[\begin{array}[]{llllll}7.4071&0.0000&7.5586&2.1030&4.2434\\\ 0.0000&0.0000&0.0000&0.0000&0.0000\\\ 7.5586&0.0000&7.7132&2.1460&4.3302\\\ 2.1030&0.0000&2.1460&0.5971&1.2048\\\ 4.2434&0.0000&4.3302&1.2048&2.4310\end{array}\right]$
$\left[\begin{array}[]{ll}0.1911\\\ 0.0504\\\ 0.5484\\\ 0.0095\\\ 0.2006\end{array}\right]$ | $\left[\begin{array}[]{ll}2.1900\\\ -1.7656\\\ -1.3755\\\ -1.7914\\\ 7.0248\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0000\\\ 1.2165\\\ 0.2253\\\ 0.4747\\\ 0.7783\end{array}\right]$ | $\left[\begin{array}[]{lllll}0.0000&0.0000&0.0000&0.0000&0.0000\\\ 0.0000&1.4799&0.2741&0.5775&0.9469\\\ 0.0000&0.2741&0.0508&0.1070&0.1753\\\ 0.0000&0.5775&0.1070&0.2254&0.3695\\\ 0.0000&0.9469&0.1753&0.3695&0.6058\end{array}\right]$
$\left[\begin{array}[]{ll}0.2166\\\ 0.6291\\\ 0.0149\\\ 0.0433\\\ 0.0961\end{array}\right]$ | $\left[\begin{array}[]{ll}3.5136\\\ -9.2542\\\ 9.9301\\\ 6.7545\\\ 0.6759\end{array}\right]$ | $\left[\begin{array}[]{ll}2.7216\\\ 0.0000\\\ 2.7773\\\ 0.7727\\\ 1.5592\end{array}\right]$ | $\left[\begin{array}[]{lllll}7.4071&0.0000&7.5586&2.1030&4.2434\\\ 0.0000&0.0000&0.0000&0.0000&0.0000\\\ 7.5586&0.0000&7.7132&2.1460&4.3302\\\ 2.1030&0.0000&2.1460&0.5971&1.2048\\\ 4.2434&0.0000&4.3302&1.2048&2.4310\end{array}\right]$
$\left[\begin{array}[]{ll}0.2583\\\ 0.0041\\\ 0.0905\\\ 0.5958\\\ 0.0513\end{array}\right]$ | $\left[\begin{array}[]{ll}2.2080\\\ -1.8738\\\ -1.0651\\\ -1.9303\\\ 6.8716\end{array}\right]$ | $\left[\begin{array}[]{ll}0.0857\\\ 1.1782\\\ 0.3057\\\ 0.4841\\\ 0.8029\end{array}\right]$ | $\left[\begin{array}[]{lllll}0.0073&0.1010&0.0262&0.0415&0.0688\\\ 0.1010&1.3882&0.3601&0.5704&0.9460\\\ 0.0262&0.3601&0.0934&0.1480&0.2454\\\ 0.0415&0.5704&0.1480&0.2344&0.3887\\\ 0.0688&0.9460&0.2454&0.3887&0.6447\end{array}\right]$

Note that we can verify that problem (SOQP) is nonconvex with seven different
choices of weighted coefficient $\lambda$ which show in Table 7. So, Example 4
can be solved by using the proposed method. The results of $x^{*}$ and $X^{*}$
in Table 7 show that $X^{*}=x^{*}(x^{*})^{T}$ holds for these seven different
cases of weighted coefficient $\lambda$. Hence, we can conclude that each
$x^{*}$ in Table 7 also is Pareto optimal solution for Example 4. Furthermore,
compare the results of FV and $x^{*}$ in Tables 6 and 7, note that some weakly
Pareto optimal solutions of Example 4 can be obtained by using our method. For
example, if $\lambda=(0.3317,0.1522,0.3480,0.1217,0.0464)$, then we obtain
weakly Pareto optimal solution $(0.0000,1.2165,0.2253,0.4747,0.7783)^{T}$ for
Example 4.

######  Remark 3.

The results for Example 3 and Example 4 imply that we not only obtain Pareto
optimal solutions, but also obtain some weakly Pareto optimal solutions for
original problem with appropriate choices of weighted coefficient $\lambda$.
Summarizing these results, we can conclude that our method is effective for
solving some problems (MOQP).

######  Example 5.

(Portfolio Selection Problem) This problem is taken from [16]. It is a
practical portfolio selection problem in which objective function $F(x)$ has
the following expression

$F(x)=(\frac{1}{2}x^{T}Q_{1}x,\ -c_{2}^{T}x,\ -c_{3}^{T}x)^{T},$

where symmetric matrix $Q_{1}$ is called the risk matrix, $c_{2}$ denotes the
return rate vector, $c_{3}$ is a given weighting vector and its element is a
function of corresponding security liquidity. The corresponding coefficients
are given in Table 8.

Table 8. Coefficients for Example 5
$Q_{1}=\left[\begin{array}[]{llllllllllllllllllllll}12.35&-8.64&9.88&-8.43&9.42&-1.47&4.29&-0.96&15.2&0\\\
-8.64&6.42&-8.15&5.13&4.56&5.83&-6.72&0.78&3.42&0\\\
9.88&-8.15&18.27&-8.14&2.32&-4.12&-1.85&1.34&0.795&0\\\
-8.43&5.13&-8.14&12.6&-8.43&7.47&-4.67&-0.679&8.09&0\\\
9.42&4.56&2.32&-8.43&17.07&7.33&-3.18&-0.547&0.323&0\\\
-1.47&5.83&-4.12&7.47&7.33&13.38&-3.84&-0.478&0.616&0\\\
4.29&-6.27&-1.85&-4.67&-3.18&-3.84&3.41&0.107&0.464&0\\\
-0.96&-0.78&1.34&-0.679&-0.547&-0.478&0.107&1.33&11.23&0\\\
15.2&3.42&0.795&0.809&0.323&0.616&0.464&11.23&2.45&0\\\
0&0&0&0&0&0&0&0&0&0\end{array}\right]$
---
$c_{2}=[11.11\ \ 17.78\ \ 2.22\ \ 18.28\ \ 12.6\ \ 10.15\ \ 6.72\ \ 5.75\ \
6.1\ \ 2.25]^{T}$
$c_{3}=[0.069\ \ 0.092\ \ 0.046\ \ 0.096\ \ 0.086\ \ 0.068\ \ 0.06\ \ 0.145\ \
0.151\ \ 0.197]^{T}$
$A=[1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1\ 1]\ \ \ b=1$

Note that the objective functions contain only one quadratic function
$\frac{1}{2}x^{T}Q_{1}x$, by using eig function of MATLAB, it is easy to
verify that function $\frac{1}{2}x^{T}Q_{1}x$ is nonconvex. Thus, we use the
proposed method to solve this problem. The corresponding optimal results are
given in Table 9.

Table 9. Optimal numerical results for Example 5 $\lambda$ | FV | $x^{*}$
---|---|---
$\left[\begin{array}[]{ll}0.3482\\\ 0.1655\\\ 0.4863\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1894\\\ -15.6800\\\ -0.0850\end{array}\right]$ | $(0.3240,0.5567,0.0000,0.1194,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000)^{T}$
$\left[\begin{array}[]{ll}0.5141\\\ 0.1931\\\ 0.2927\end{array}\right]$ | $\left[\begin{array}[]{ll}0.1320\\\ -15.5462\\\ -0.0846\end{array}\right]$ | $(0.3433,0.5466,0.0000,0.1101,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000)^{T}$
$\left[\begin{array}[]{ll}0.6167\\\ 0.3087\\\ 0.0746\end{array}\right]$ | $\left[\begin{array}[]{ll}0.2026\\\ -15.7070\\\ -0.0851\end{array}\right]$ | $(0.3200,0.5590,0.0000,0.1210,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000)^{T}$

In Table 9, Example 5 is solved with three different values of weight
$\lambda$. The first element and the absolute value of the second element of
FV in Table 9 denote the expectation risk and return, respectively. The
results in Table 9 show that we obtain the lower risk and higher return when
weight $\lambda$ is chosen appropriately. Moreover, we also compare with the
results in [16], which are show in Table 10.

Table 10. Optimal numerical results for Example 5 in [16] FV | $x^{*}$
---|---
$\left[\begin{array}[]{ll}0.6823\\\ -7.5079\\\ -0.1343\end{array}\right]$ | $(0.0453,0.0706,0.0273,0.0703,0.078,0.045,0.0534,0.169,0.1665,0.2746)^{T}$

The results of FV in Tables 9 further imply that the risk and return are more
comparable with the results in Table 10. For example, when
$\lambda=(0.3482,0.1655,0.4863)$, we obtain lower risk $0.1894$ and higher
return $15.6800$, which are more comparable with the results of [16] $0.6823$
and $7.5079$, respectively. Furthermore, note that the optimal solutions
$x^{*}$ obtained by using our method are sparse, which imply that we mainly
focus on some kinds of important stocks. Hence, we can put together the
limited money, and invest these money in some of important stocks to obtain
more satisfied return with lower risk.

We also notice that there are infinite choices of weight $\lambda$, and not
all choices of weight $\lambda$ are reasonable for each investor. How to
select weight $\lambda$ to lower risk and higher return depends on the
investors’s preference. Therefore, it is reasonable that investors participate
in decision making and continuously revise their preferences according to
practical conditions. This also shows that our method is promising in solving
portfolio selection problems.

##  6\. Concluding remarks

In this paper, a class of (MOQP) problems is discussed. By using the linear
weighted sum method to deal with quadratic objective functions, problem (MOQP)
is transformed into problem (SOQP), which is nonconvex in general. Then,
taking advantage of lifting techniques, problem (SOQP) is equivalently
reformulated as problem (CP) which is a convex programming problem but NP-hard
in general. A computable relaxed convex problem (DNNP) for problem (CP) is
obtained by using doubly nonnegative relaxation method. Moreover, based on the
characteristics of optimal solutions for problem (DNNP), a sufficient
condition for (weakly) Pareto optimal solutions for problem (MOQP) is
proposed. Finally, the numerical results of two problems and a practical
portfolio selection problem show that the proposed method is effective and
promising.

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Received xxxx 20xx; revised xxxx 20xx.