Robust convex quadratically constrained programs

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as second-order cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural. 1 Problem formulation A generic quadratically constrained program (QCP) is defined as follows. minimize cx subject to xQix+ 2q T i x+ γi ≤ 0, i = 1, . . . , p, (1) where the vector of decision variables x ∈ R, and the data c ∈ R, γi ∈ R, qi ∈ R and Qi ∈ Rn×n, for all i = 1, . . . , p. Note that without any loss of generality one may assume that the objective is linear. The QCP (1) is a convex optimization problem if and only if Qi o 0 for all i = 1, . . . , p, where Q o 0 denotes that the matrix Q is positive semidefinite. Suppose Q o 0. Then Q = VV for some V ∈ Rm×n and the quadratic constraint xQx ≤ −(2qTx+ γ) is equivalent to the second-order cone (SOC) constraint [2, 18, 22] ∥∥∥∥ [ 2Vx (1 + γ + 2qx) ]∥∥∥∥ ≤ 1− γ − 2q x. (2) Submitted to Math Programming, Series B. Please do not circulate. IEOR Department, Columbia University, Email: [email protected]. Research partially supported by DOE grant GE-FG01-92ER-25126, and NSF grants DMS-94-14438, CDA-97-26385 and DMS-01-04282 IEOR Department, Columbia University, Email: [email protected]. Research partially supported by NSF grants CCR-00-09972 and DMS-01-04282.