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SDP reformulation for robust optimization problems based on nonconvex QP duality

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Abstract

In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvex quadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with “single” (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand’s Lorentz positivity.

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Notes

  1. Notice that the single ellipsoidal uncertainty is considered for each SOC constraint. Therefore, if the SOCP has K SOC constraints, then the whole uncertainty set consists of K independent ellipsoids. (See Sect. 4.)

  2. Typically, Ω is ℝn, \(\mathbb {R}^{n}_{+}\), or a polyhedral set characterized by a finite number of linear equalities and inequalities.

  3. In general, set Z can depends on the true value of Player 2’s strategy. For more details, see [23].

  4. The robust Nash equilibrium coincides with the well-known Nash equilibrium when Y={y}, Z={z}, D A ={A} and D B ={B}.

  5. By the constraints of SDP (3.16), \(P^{i}_{0}(x^{*}) - \alpha_{i}^{*} P^{i}_{1} - \beta_{i}^{*} P^{i}_{2} \succeq0\) always holds at the optimum \((x^{*}, \alpha^{*}, \beta^{*}, \lambda_{0}^{*})\).

  6. If m i =1 for some i, then the constraint can be rewritten as two linear inequalities \(-(\hat{c}^{i})^{\top}x + \hat{d}^{i} \le\hat{A}^{i}x + \hat{b}^{i} \le(\hat{c}^{i})^{\top}x + \hat{d}^{i}\). So existing frameworks can be applied. (See Ben-Tal and Nemirovski [9].)

  7. The problem where \((\hat{A}, \hat{b}, \hat{c}, \hat{d})\) is replaced by (A 0,b 0,c 0,d 0) is called a nominal problem.

  8. Note that, if a nominal problem has an optimal solution, then the objective function value of problem (5.1) is bounded below. (The feasible region of problem (5.1) becomes smaller as κ increases.)

  9. Actually, we did not apply the Hildebrand-based approach to instances satisfying condition (4.13), since the RC optimality of our approach is theoretically guaranteed for such problem instances.

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Authors and Affiliations

  1. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Ryoichi Nishimura, Shunsuke Hayashi & Masao Fukushima

Authors
  1. Ryoichi Nishimura
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  2. Shunsuke Hayashi
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  3. Masao Fukushima
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Correspondence to Shunsuke Hayashi.

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This research was supported in part by Grant-in-Aid for Young Scientists (B) and Scientific Research (C) from Japan Society for the Promotion of Science.

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Nishimura, R., Hayashi, S. & Fukushima, M. SDP reformulation for robust optimization problems based on nonconvex QP duality. Comput Optim Appl 55, 21–47 (2013). https://doi.org/10.1007/s10589-012-9520-9

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