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.

Robust sensitivity analysis of the optimal value of linear programming

“…The computation times for computing r − and r + are trivial, while the average computation time for computing v − and v + is about 77 seconds. From the results in Table 2, we see clearly that our approach recovers q + for all 10 instances, which also matches the quality of results from [29].…”

“…Generally speaking, LP-based relaxations are relatively weak (see [1]); we do not consider them in this paper. In addition, SDP approaches can often be tailored to outperform the more general Lasserre approach as has been demonstrated in [29]. Our copostive-and SDP-based approach is similar; see for example the valid inequalities discussed in Section 3.2.…”

“…We next consider an example for calculating systemic risk in financial systems, which is an application of worst-case linear optimization (WCLO) presented in [29].…”

“…For an interbank market, systemic risk is used to evaluate the potential loss of the whole market as a response to the decisions made by the individual banks [13,29]. Specifically, let us consider a market consisting of n banks.…”

“…In practice, however, there exist uncertainties in the exogenous operating cash flows. Allowing for uncertainties, the worst-case systemic risk problem [29] is given as [29], we randomly generate 10 instances of size m × n = 3 × 5. In accordance with Remark 1, which states that q − is easy to calculate in this case, we focus our attention on the worst-case value q + .…”