Adjustable robust multiobjective linear optimization: Pareto optimal solutions via conic programming

Abstract

In this paper we study two-stage affinely adjustable robust multi-objective optimization problems. We show how (weak) Pareto optimal solutions of these robust multi-objective problems can be found by solving conic linear programming problems. We do this by first deriving numerically verifiable conditions that characterize (weak) Pareto optimal solutions of affinely adjustable robust multi-objective programs under a spectrahedron uncertainty set. The uncertainty set covers most of the commonly used uncertainty sets of robust optimization. We then reformulate the weighted-sum optimization problems of the multi-objective problems, derived with the aid of the optimality conditions, as equivalent conic linear programming problems, such as semidefinite programs or second-order cone programs, to find the (weak) Pareto optimal solutions. We illustrate by an example how our results can be used to find a second-stage (weak) Pareto optimal solution by solving a semidefinite program using a commonly available software.

Appendix: Robust multi-objective linear programs and SDP reformulations

In this appendix, we give optimality conditions and semidefinite programming reformulations for a single-stage robust multi-objective linear optimization models that can be used to transform the previous two-stage optimization problem to a single-stage problem.

Let us consider a robust counterpart of (UP) given by

where $f_r(x):=\max \limits _{u\in U}c_r(u)^\top x, r=1,\dots ,p$ for $x\in {\mathbb {R}}^{n}.$

The following lemma presents optimality conditions of the robust multi-objective program (RP) that has been employed in the proofs of the previous sections.

Lemma 4.1

(Necessary optimality for (RP)) For the problem  (RP), let the cone C be closed, where

$$\begin{aligned} C:=\textrm{coneco}\big \{\big (a_j(u), b_j(u) \big ), j=1,\ldots ,q\mid u:=(u_1,\ldots ,u_s)\in U\big \}\subset {\mathbb {R}}^{n+1}.\end{aligned}$$

(5.11)

If $\bar{x}\in {\mathbb {R}}^{n}$ is a weak Pareto solution of problem (RP), then, there exist $(\alpha _1,\ldots ,\alpha _p)\in {\mathbb {R}}^p_+\setminus \{0\}$, $\lambda _j\ge 0, j= 1,\ldots ,q, \mu \ge 0$, ${{\bar{u}}}^r \in U, r=1,\ldots ,p$ and $u^j\in U, j= 1,\ldots ,q,$ such that

$$\begin{aligned}&\sum _{r=1}^p\alpha _rc_r({{\bar{u}}}^r)+\sum \limits _{j=1}^q\lambda _ja_j(u^j)=0_n,\quad \sum \limits _{j=1}^q\lambda _jb_j(u^j)+\sum _{r=1}^p \alpha _rf_r(\bar{x})+\mu =0.\end{aligned}$$

(5.12)

Proof

The proof uses a classical alternative theorem (see e.g., (Rockafellar 1970, Theorem 21.1) to show that every weak Pareto solution can be determined by a convex program  (RP) (cf. Ehrgott (2005)). Applying a classical minimax theorem (see e.g., ( Sion 1958, Theorem 4.2) and a strong separation theorem (see e.g., Mordukhovich & Nam 2014, Theorem 2.2) gives the desired result. We omit the details for the purpose of a shorter presentation the paper. $\square$

For each $\alpha :=(\alpha _1,\ldots ,\alpha _p)\in {\mathbb {R}}^p_+\setminus \{0\},$ we consider a (scalar) robust weighted-sum optimization problem of (RP) as follows:

A semidefinite programming reformulation problem for (RP$_{\alpha }$) is given by

The following lemma presents an exact semidefinite programming (SDP) reformulation for (RP$_{\alpha }$) that has been employed in the proofs of the previous sections.

Lemma 4.2

(Exact SDP reformulation for (RP$_{\alpha }$)) Let $\alpha :=(\alpha _1,\ldots ,\alpha _p)\in {\mathbb {R}}^p_+\setminus \{0\}$ be such that the problem (RP$_{\alpha }$) admits an optimal solution and assume that ${{\bar{x}}}\in {\mathbb {R}}^{n}$ is an optimal solution of problem (RP$_{\alpha }$). Assume that the cone C in (5.11) is closed and there exists ${{\hat{u}}}:= ({{\hat{u}}}_1,\ldots ,{{\hat{u}}}_s)\in {\mathbb {R}}^s$ such that $A_0+\sum _{i=1}^s{{\hat{u}}}_iA_i\succ 0$. Then, there exist $\tilde{Z}_r\succeq 0, r=1,\ldots ,p, Z_j\succeq 0, j=1,\ldots ,q$ such that $({{\bar{x}}},{{\tilde{Z}}}_1,\ldots ,{{\tilde{Z}}}_p,Z_1,\ldots , Z_q)$ is an optimal solution of problem (RP$^*_{\alpha }$) and

$$\begin{aligned} \textrm{val}(RP_{\alpha })= \textrm{val} (RP^{*}_{\alpha })={\sum _{r=1}}^p{\alpha _{r}}\big ({c_{r}}^{0\top }{{\bar{x}}} +\textrm{Tr}({A_{0}}{{\tilde{Z}}}_{r})\big ), \end{aligned}$$

(4.13)

where val(RP$_{\alpha }$) and val(RP$^*_{\alpha }$) stand for the optimal values of problem (RP$_{\alpha }$) and problem (RP$^*_{\alpha }$), respectively.

Proof

The proof uses a strong duality in semidefinite programming (cf. Blekherman 2012, Theorem 2.15) to get the desired result. $\square$