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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    • "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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    • "In literatures, two similar problems are also discussed: in [6] [7] [8] the convexity of the quadratic constraint is removed, while in [9] [10] the constraint is replaced by a two-sided (lower and upper bounded) quadratic constraint. Although the function P(x) may be nonconvex, it is proved that the problem (P) possesses the hidden convexity, i.e. (P) is actually equivalent to a convex optimization problem [10], and for each optimal solution x, there exists a Lagrange multiplier m such that the following conditions hold [11]: "
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    • "Finally, there have been prior observations regarding the presence of hidden convexity in nonconvex programs (cf. [4]). 4. General uncertain non-monotone LCPs. "
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