A nonlinear semidefinite optimization relaxation for the worst-case linear optimization under uncertainties

Abstract

In this paper, we consider the so-called worst-case linear optimization (WCLO) with uncertainties in the right-hand-side of the constraints. Such a problem arises from numerous applications such as systemic risk estimate in finance and stochastic optimization. We first show the problem is NP-hard and present a coarse semidefinite relaxation (SDR) for WCLO. An iterative procedure is introduced to sequentially refine the relaxation model based on the solution of the current relaxation model by simply changing some parameters in the coarse SDR. We show that the sequence of the proposed SDRs will converge to some nonlinear semidefinite optimization problem (SDO). A bi-section search algorithm is proposed to solve the resulting nonlinear SDO. Our preliminary experimental results illustrate that the nonlinear SDR can provide very tight bounds for the original WCLO and is able to locate the exact global solution in most cases within a few iterations on average.

Appendices

Appendix A: Proof of Proposition 1.1

Using the strong duality of linear program, we have

$$\begin{aligned}&\max \limits _{\left\Vert u \right\Vert _2\le 1} {\min \limits _{Ax\le Qu + b_0}} ~~~c^T x\end{aligned}$$

(25)

$$\begin{aligned}&\quad = \max \limits _{\left\Vert u \right\Vert _2\le 1} {\max \limits _{A^T y = c,~y\le 0}} ~~~u^T Q^T y + b_0^T y\end{aligned}$$

(26)

$$\begin{aligned}&\quad = \max \limits _{\left\Vert u \right\Vert _2\le 1} {\max \limits _{A^T y = c,~y\le 0}} ~~~\left\Vert u \right\Vert _2 \left\Vert Q^T y \right\Vert _2 + b_0^T y\end{aligned}$$

(27)

$$\begin{aligned}&\quad = \max \limits _{A^T y = c,~y\le 0} ~~~\left\Vert Q^T y \right\Vert _2 + b_0^T y. \end{aligned}$$

(28)

With carefully chosen $(Q, b_0, A)$, the above optimization problem includes the following problem as a special case,

$$\begin{aligned}&\max \limits _{x}&\left\Vert x \right\Vert _2\end{aligned}$$

(29)

$$\begin{aligned}&\text {s.t.}&Bx\ge d \end{aligned}$$

(30)

which has been shown to be NP-hard by a reduction to the NP-complete partition problem [18, 25].

Appendix B: Affine-rule approximation

A common tool to approximately solve multilevel optimization including WCLO is using the affine-rule approximation (see e.g., [3, 4, 7, 9, 10, 13]). More specifically, for WCLO, we have

$$\begin{aligned} \max \limits _{\left\Vert u \right\Vert _2\le 1} ~~~{ \min \limits _{{ \begin{array}{c}Ax\le Qu + b_0\end{array}}} } ~~~c^T x = \min \limits _{{ \begin{array}{c} Ax(u)\le Qu + b_0,\\ \forall \left\Vert u \right\Vert _2\le 1\end{array}}} ~~~\max \limits _{\left\Vert u \right\Vert _2\le 1} ~~~c^T x(u) \end{aligned}$$

(31)

since we can always choose

$$\begin{aligned} x(u)=\arg \min \limits _{{ \begin{array}{c}Ax\le Qu + b_0\end{array}}} c^T x. \end{aligned}$$

To make problem (31) tractable, we can artificially restrict the decision function $x(\cdot )$ to be affine in uncertainties, i.e.,

$$\begin{aligned} x(u) = Pu+q. \end{aligned}$$

Under such a circumstance, problem (31) reduces to

$$\begin{aligned}&\min \limits _{{ \begin{array}{c} A(Pu+q)\le Qu + b_0,\\ \forall \left\Vert u \right\Vert _2\le 1\end{array}}} ~~~\max \limits _{\left\Vert u \right\Vert _2\le 1} ~~~c^T (Pu+q) \nonumber \\&\quad = \begin{array}{ccc} \min \limits _{P,q,t} &{} t &{} \nonumber \\ {\text {s.t.}} &{} A(Pu+q)\le Qu + b_0, &{}\forall \left\Vert u \right\Vert _2\le 1\\ &{} c^T (Pu+q)\le t, &{} \forall \left\Vert u \right\Vert _2\le 1 \end{array}\\&\quad = \begin{array}{cl} \min \limits _{P,q,t} &{} t \\ {\text {s.t}} &{} \left\Vert A_i P - Q_i \right\Vert _2 + A_i q\le b_{0_i},~i = 1,\ldots ,m\\ &{} \left\Vert P^T c \right\Vert _2 + c^T q\le t, \end{array} \end{aligned}$$

(32)

which is a tractable second-order conic optimization problem.