Hidden convexity in some nonconvex quadratically constrained quadratic programming

Mathematical Programming (Impact Factor: 1.8). 01/1996; 72(1):51-63. DOI: 10.1007/BF02592331
Source: DBLP


We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic
constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems),
we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then
consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints,
which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations
which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special
cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent
convex programs.

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Available from: Aharon Ben-Tal, Feb 10, 2015
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    • "For the primal problem (1), the primal Slater condition requires an x 0 such that 1 2 x T 0 Bx 0 − g T x 0 < μ. Ben-Tal and Teboulle [1] and Xing et al. [22] needed the dual Slater condition (A1) whereas Ye and Zhang's exact semi-definite programming approach [23] did both. Although (A2)+(A3) is weaker than (A1), they can be reduced to (A1) after space reduction. "

    Full-text · Dataset · Feb 2016
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    • "It is not well understood when one can make such convex transformations in variational problems; but, when one can, it always simplifies things since we have immediate access to Theorem 1.2.7, and need only verify that the first-order necessary condition holds. Especially for hidden convexity in quadratic programming there is substantial recent work, see e.g.[50,440]. "
    Dataset: sample

    Full-text · Dataset · Jan 2016
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    • "instance, allowed the authors in [6] to re-write the original problem in a more tractable one. The symbol LD stands for linear dependence. "
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    ABSTRACT: We establish various extensions of the convexity Dines theorem for a (joint-range) pair of inhomogeneous quadratic functions. If convexity fails we describe those rays for which the sum of the joint-range and the ray is convex. These results are suitable for dealing nonconvex inhomogeneous quadratic optimization problems under one quadratic equality constraint. As applications of our main results, different sufficient conditions for the validity of S-lemma (a nonstrict version of Finsler's theorem) for inhomogenoeus quadratic functions, is presented. In addition, a new characterization of strong duality under Slater-type condition is established.
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