Robust Pareto solutions for convex quadratic multiobjective optimization problems under data uncertainty

Abstract

In this paper, we consider a convex quadratic multiobjective optimization problem, where both the objective and constraint functions involve data uncertainty. We employ a deterministic approach to examine robust optimality conditions and find robust (weak) Pareto solutions of the underlying uncertain multiobjective problem. We first present new necessary and sufficient conditions in terms of linear matrix inequalities for robust (weak) Pareto optimality of the multiobjective optimization problem. We then show that the obtained optimality conditions can be alternatively checked via other verifiable criteria including a robust Karush–Kuhn–Tucker condition. Moreover, we establish that a (scalar) relaxation problem of a robust weighted-sum optimization program of the multiobjective problem can be solved by using a semidefinite programming (SDP) problem. This provides us with a way to numerically calculate a robust (weak) Pareto solution of the uncertain multiobjective problem as an SDP problem that can be implemented using, e.g., MATLAB.

Appendix: Numerical examples

The following examples show how we can employ the SDP reformulation scheme obtained in Theorem 4.1 (Procedure 1) to find robust (weak) Pareto solutions of uncertain convex quadratic multiobjective optimization problems.

The first example is dealt with a family of uncertain convex quadratic multiobjective problems of ($\hbox {EU}_c$) by specifying the value c of the objectives, while the second example is concerned with a family of uncertain convex quadratic multiobjective problems of (EU$_n$) by specifying the dimension n of the decision variables.

Example 6.1

(Calculating Pareto solutions via SDP) Let us consider a family of uncertain convex quadratic multiobjective problems:

where $c\in [0,+\infty )$ is a given parameter, $u:=(u_1,u_2)\in \Omega$ and $v:=(v_1,v_2)\in \Theta$ are uncertain parameters and $f_r, r=1,2,3,4$ and $g_l, l=1,2,3,4,5,6$ are bi-functions given by

$$\begin{aligned} f_1(x,u)&:=x_1^2+x_2^2-cx_1+1+u_1+u_2,\; f_2(x,u):=x^2_2+1-u_1+u_2,\\ f_3(x,u)&:=x_2^2+2x_2+u_1,\;f_4(x,u):=x_1^2+4x_2^2-cx_1+x_2-u_2,\\ g_1(x,v)&:=x_2^2-x_1+v_1x_2+v_2x_1-2c-4,\; g_2(x,v):=-x_1+v_1-3,\\ g_3(x,v)&:=-x_2-v_2-3,\;g_4(x,v):=x^2_1+x_2^2-v_1x_1-v_2x_2-c^2-2c-5,\\ g_5(x,v)&:=x_2-4,\;g_6(x,v):=x_1-c-4,\;x:=(x_1,x_2)\in \mathbb {R}^2. \end{aligned}$$

Here, the uncertainty sets $\Omega$ and $\Theta$ are given by

$$\begin{aligned} \Omega :=\{u:=(u_1,u_2)\in \mathbb {R}^2\mid u_1^2+ u_2^2\le 1\},\; \Theta :=\{v:=(v_1,v_2)\in \mathbb {R}^2\mid \frac{v_1^2}{4}+\frac{v^2_2}{9}\le 1\}. \end{aligned}$$

(6.1)

Now, we consider the robust convex quadratic multiobjective problem of ($\hbox {EU}_c$) as follows:

Note that the problem ($\hbox {ER}_c$) can be expressed in terms of problem (RP), where $\Omega _r:=\Omega , r=1,2,3,4,$ $\Theta _l:=\Theta , l=1,2,3,4,5,6$ are described respectively by

$$\begin{aligned} A^r&:=I_3, B^l:= \left( \begin{array}{ccc} 4 &{} 0 &{}0 \\ 0&{}9&{}0\\ 0&{}0&{}1\\ \end{array} \right) , A^r_1:=B^l_1= \left( \begin{array}{ccc} 0 &{} 0 &{}1 \\ 0&{}0&{}0\\ 1&{}0&{}0\\ \end{array} \right) , A^r_2:=B^l_2= \left( \begin{array}{ccc} 0 &{} 0 &{}0 \\ 0&{}0&{}1\\ 0&{}1&{}0\\ \end{array} \right) \end{aligned}$$

and $f_r, r=1,2,3,4, g_l, l=1,2,3,4,5,6$ are given respectively by $Q^1:=I_2, Q^2:=Q^3:=\left( \begin{array}{cc} 0 &{} 0 \\ 0&{}1\\ \end{array} \right) , Q^4:=\left( \begin{array}{cc} 1 &{} 0 \\ 0&{}4\\ \end{array} \right) ,$ $\xi ^1:=(-c,0), \xi ^2:=0_2, \xi ^3:=(0,2), \xi ^4:=(-c,1), \xi ^r_i:=0_2, r=1,2,3,4, i=1,2, \beta ^1:=\beta ^2:=1, \beta ^3:=\beta ^4:=0, \beta ^1_1:=\beta ^1_2:=\beta ^2_2:=\beta ^3_1:=1, \beta ^2_1:=\beta ^4_2:=-1, \beta ^3_2:=\beta ^4_1:=0$ and $M^1:=\left( \begin{array}{cc} 0 &{} 0 \\ 0&{}1\\ \end{array} \right) ,$ $M^2:=M^3:=M^5:=M^6:=0_{2\times 2}$, $M^4:=I_2, \theta ^1:=\theta ^2:=(-1,0)$, $\theta ^3:=(0,-1), \theta ^4:=0_2, \theta ^5:=(0,1)$, $\theta ^6:=(1,0)$, $\theta ^1_1:=(0,1), \theta ^1_2:=(1,0)$, $\theta ^2_1:=\theta ^2_2$ $:=\theta ^3_1:=\theta ^3_2:=\theta ^5_1:=\theta ^5_2$ $:=\theta ^6_1:=\theta ^6_2:=0_2, \theta ^4_1:=(-1,0)$, $\theta ^4_2=(0,-1), \gamma ^1:=-2c-4$, $\gamma ^2:=\gamma ^3:=-3, \gamma ^4:=-c^2-2c-5$, $\gamma ^5:=-4, \gamma ^6:=-c-4, \gamma ^1_1:=\gamma ^1_2$ $:=\gamma ^2_2:=\gamma ^3_1$ $:=\gamma ^4_1:=\gamma ^4_2:=\gamma ^5_1$ $:=\gamma ^5_2:=\gamma ^6_1:=\gamma ^6_2:=0$, $\gamma ^2_1:=1,\gamma ^3_2:=-1.$

We now use the SDP reformulation, obtained in Theorem 4.1, to find a robust (weak) Pareto solution of problem ($\hbox {EU}_c$). Taking $\tilde{x}:=(c,1),$ we see that $g_l(\tilde{x},v) <0$ for all $v\in \Theta$ and $l=1,2,3,4,5,6.$ Thus, the characteristic cone $K:={\text {coneco}} \{ (0_2,1)\cup \mathrm{epi}\,g^*_l(\cdot ,v), v\in \Theta , l=1,2,3,4,5,6\}$ is closed by virtue of Remark 3.2. Moreover, by taking $\tilde{u}:=\tilde{v}:=(0,0)\in \mathbb {R}^2,$ it holds that $A^r+\sum \limits _{i=1}^2\tilde{u}_iA^r_i\succ 0, r=1,2,3,4$ and $B^l+\sum \limits _{j=1}^2\tilde{v}_jB^l_j\succ 0, l=1,2,3,4,5,6$. Namely, all the assumptions of Theorem 4.1 are fulfilled in this setting.

Let $\alpha :=(1,1,1,1)\in \mathrm{int}\mathbb {R}^4_+,$ and consider a corresponding robust weighted-sum optimization problem of ($\hbox {ER}_c$) as follows:

Note that the (scalar) weighted-sum problem (E$_c$) has optimal solutions as its objective function is a continuous function and its feasible set is a compact set.

The SDP reformulation problem of (E$_c$) is given by

Using the Matlab toolbox CVX (see e.g., Grant and Boyd, 2014), we solve the problem (E$^{*}_c$) with (for instance) $c:=4$ and the solver returns the weighted-sum optimal value as $-1.17157$ and an optimal solution of problem (E$^{*}_c$) with $c:=4$ as $(\bar{w}, \bar{W}, \bar{\lambda }^r, \bar{t}^l, \tilde{Z}^r_0, Z^l_0, r=1,2,3,4, l=1,2,3,4,5,6)$, where $\bar{w}=(2.0000, -2.9400\text {e-}08)\approx (2,0)$. By Theorem 4.1(iii), we conclude that $\bar{w}=(2, 0)$ is a robust (weak) Pareto solution of problem ($\hbox {EU}_c$) with $c:=4$. (In this setting, we can re-check directly that $\bar{w}$ is a (weak) Pareto solution of problem ($\hbox {ER}_c$) with $c:=4$.)

Similarly, we test with some other values of c (see Table 1). These the numerical tests are conducted on a computer with a 1.90GHz Intel(R) Core(TM) i7-8650U and 16.0GB RAM, equipped with MATLAB R2018b. In Table 1, “Robust Pareto Solutions” are optimal decision variables $x:=(x_1,x_2)$ of ($\hbox {EU}_c$) and “Weighted-sum Values” are optimal values of (E$^{*}_c$) with the corresponding values of c.

Table 1 Robust Pareto solutions for ($\hbox {EU}_c$) with other values of c

Example 6.2

(Calculating Pareto solutions with higher dimensional decision variables) Let us consider a family of uncertain convex quadratic multiobjective problems:

where $n\in \mathbb {N}, n\ge 3,$ is a given parameter, $u:=(u_1,u_2)\in \Omega$ and $v:=(v_1,v_2,v_3)\in \Theta$ are uncertain parameters and $f_r, r=1,2,3,4,5$ and $g_l, l=1,2,3$ are bi-functions given by

$$\begin{aligned} f_1(x,u)&:=x_1^2+x_2^2-nx_n+1+u_1-u_2,\; f_2(x,u):=x^2_2-x_1-\ldots -x_n+1-u_1+u_2,\\ f_3(x,u)&:=x_2^2+2x_2-x_n+u_1+u_2,\;f_4(x,u):=x_1^2+4x_2^2-x_1+nx_n-u_2,\\ f_5(x,u)&:=x_1+x_2+\ldots +x_n-u_1+u_2,\\ g_1(x,v)&:=x_2^2-x_1+v_1x_2+v_2x_1-2n,\; g_2(x,v):=x_1^2+x_2^2+\ldots +x^2_n-n+v_1-v_3,\\ g_3(x,v)&:=x^2_1+x_2^2-v_1x_1-v_2x_2-x_n-n+v_3,\;x\in \mathbb {R}^n. \end{aligned}$$

Here, the uncertainty sets $\Omega$ and $\Theta$ are given by

$$\begin{aligned}&\Omega :=\{u:=(u_1,u_2)\in \mathbb {R}^2\mid u_1^2+\frac{1}{2} u_2^2\le 1, u_1\ge 0\},\\&\quad \Theta :=\{v:=(v_1,v_2,v_3)\in \mathbb {R}^3\mid v_1^2+v^2_2+v^2_3\le 1\}. \end{aligned}$$

Now, we consider the robust convex quadratic multiobjective problem of (EU$_n$) as follows:

Note that the problem (ER$_n$) can be expressed in terms of problem (RP), where $\Omega _r:=\Omega , r=1,2,3,4,5,$ $\Theta _l:=\Theta , l=1,2,3$ are described respectively by

$$\begin{aligned}&A^r:= \left( \begin{array}{cccc} 1 &{} 0 &{}0&{}0 \\ 0&{}2&{}0&{}0\\ 0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0\\ \end{array} \right) , A^r_1:= \left( \begin{array}{cccc} 0 &{} 0 &{}1&{}0 \\ 0&{}0&{}0&{}0\\ 1&{}0&{}0&{}0\\ 0&{}0&{}0&{}1\\ \end{array} \right) , A^r_2:= \left( \begin{array}{cccc} 0 &{} 0 &{}0&{}0 \\ 0&{}0&{}1&{}0\\ 0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0\\ \end{array} \right) ,\\ {}&B^l:=I_4, B^l_1:= \left( \begin{array}{cccc} 0 &{} 0 &{}0&{}1 \\ 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0\\ 1&{}0&{}0&{}0\\ \end{array} \right) , B^l_2:= \left( \begin{array}{cccc} 0 &{} 0 &{}0&{}0 \\ 0&{}0&{}0&{}1\\ 0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ \end{array} \right) , B^l_3:= \left( \begin{array}{cccc} 0 &{} 0 &{}0&{}0 \\ 0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1\\ 0&{}0&{}1&{}0\\ \end{array} \right) \end{aligned}$$

and $f_r, r=1,2,3,4,5, g_l, l=1,2,3$ are given respectively by

$$\begin{aligned}&Q^1:= \left( \begin{array}{ccccc} 1 &{} 0 &{}0&{}\cdots &{}0 \\ 0&{}1&{}0&{}\cdots &{}0\\ 0&{}0&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ 0&{}0&{}0&{}0&{}0\\ \end{array} \right) , Q^2:= Q^3:= \left( \begin{array}{ccccc} 0 &{} 0 &{}0&{}\cdots &{}0 \\ 0&{}1&{}0&{}\cdots &{}0\\ 0&{}0&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ 0&{}0&{}0&{}0&{}0\\ \end{array} \right) , Q^4:= \left( \begin{array}{ccccc} 1 &{} 0 &{}0&{}\cdots &{}0 \\ 0&{}4&{}0&{}\cdots &{}0\\ 0&{}0&{}0&{}\cdots &{}0\\ \vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ 0&{}0&{}0&{}0&{}0\\ \end{array} \right) , \end{aligned}$$

$Q^5:=0_{n\times n}, \xi ^1:=(0,0,\ldots ,0,-n), \xi ^2:=(-1,-1,\ldots ,-1), \xi ^3:=(0,2,0,\ldots ,0,-1), \xi ^4:=(-1,0,\ldots ,0,n), \xi ^5:=(1,1,\ldots ,1), \xi ^r_i:=0_n, r=1,2,3,4,5, i=1,2, \beta ^1:=\beta ^2:=1, \beta ^3:=\beta ^4:=\beta ^5:=0, \beta ^1_1:=\beta ^2_2:=\beta ^3_1:=\beta ^3_2:=\beta ^5_2:=1, \beta ^1_2:=\beta ^2_1:=\beta ^4_2:=\beta ^5_1:=-1, \beta ^4_1:=0$ and $M^1:= Q_3, M^2:=I_{n}, M^3:=Q_1,$ $\theta ^1:=(-1,0,\ldots ,0), \theta ^2:=0_n, \theta ^3:=(0,\ldots ,0,-1), \theta ^1_1:=(0,1,0,\ldots ,0), \theta ^1_2:=(1,0,\ldots ,0), \theta ^1_3:=0_n, \theta ^2_1:=\theta ^2_2:=\theta ^2_3:=0_n,$ $\theta ^3_1:=(-1,0,\ldots ,0), \theta ^3_2=(0,-1,0,\ldots ,0), \theta ^3_3:=0_n, \gamma ^1:=-2n, \gamma ^2:=\gamma ^3:=-n, \gamma ^1_1:=\gamma ^1_2:=\gamma ^1_3:=\gamma ^2_2:=\gamma ^3_1:=\gamma ^3_2:=0,$ $\gamma ^2_1:=\gamma ^3_3:=1,\gamma ^2_3:=-1.$

We now use the SDP reformulation, obtained in Theorem 4.1, to find a robust (weak) Pareto solution of problem (EU$_n$). Taking $\tilde{x}:=0_n,$ we see that $g_l(\tilde{x},v) <0$ for all $v\in \Theta$ and $l=1,2,3.$ Thus, the characteristic cone $K:={\text {coneco}} \{ (0_n,1)\cup \mathrm{epi}\,g^*_l(\cdot ,v), v\in \Theta , l=1,2,3\}$ is closed by virtue of Remark 3.2. Moreover, by taking $\tilde{u}:=(\frac{1}{2},0)\in \mathbb {R}^2, \tilde{v}:=0_3\in \mathbb {R}^3,$ it holds that $A^r+\sum \limits _{i=1}^2\tilde{u}_iA^r_i\succ 0, r=1,2,3,4,5$ and $B^l+\sum \limits _{j=1}^3\tilde{v}_jB^l_j\succ 0, l=1,2,3$. Namely, all the assumptions of Theorem 4.1 are fulfilled in this setting.

Let $\alpha :=(1,1,1,1,1)\in \mathrm{int}\mathbb {R}^5_+,$ and consider a corresponding robust weighted-sum optimization problem of (ER$_n$) as follows:

Note that the (scalar) weighted-sum problem (E$_n$) has optimal solutions as its objective function is a continuous function and its feasible set is a compact set.

The SDP reformulation problem of (E$_n$) is given by

Using the Matlab toolbox CVX (see e.g., Grant and Boyd, 2014), we solve the problem (E$^{*}_n$) with (for instance) $n:=5$ and the solver returns the weighted-sum optimal value as $+7.56512$ and an optimal solution of problem (E$^{*}_n$) with $n:=5$ as $(\bar{w}, \bar{W}, \bar{\lambda }^r, \bar{t}^l, \tilde{Z}^r_0, Z^l_0, r=1,2,3,4,5, l=1,2,3)$, where $\bar{w}=(0.2206, -0.1376, 0, 0, 1.8757)$. By Theorem 4.1(iii), we conclude that $\bar{w}=(0.2206, -0.1376, 0, 0, 1.8757)$ is a robust (weak) Pareto solution of problem (EU$_n$) with $n:=5$. (In this setting, we can re-check directly that $\bar{w}$ is a (weak) Pareto solution of problem (ER$_n$) with $n:=5$.)

Similarly, we test with some other values of n (see Table 2). These the numerical tests are conducted on a computer with a 1.90GHz Intel(R) Core(TM) i7-8650U and 16.0GB RAM, equipped with MATLAB R2018b. In Table 2, “Robust Pareto Solutions” are optimal decision variables $x:=(x_1,\ldots , x_n)$ of (EU$_n$) and “Weighted-sum Values” are optimal values of (E$^{*}_n$) with the corresponding values of n.

Table 2 Robust Pareto solutions for (EU$_n$) with other values of n