Hidden convexity in some nonconvex quadratically constrained quadratic programming.

Mathematical Programming (Impact Factor: 2.09). 01/1996; 72:51-63. DOI:10.1007/BF02592331
Source: DBLP

ABSTRACT 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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    ABSTRACT: The paper identifies classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some partial separable structure. Among other things, the new hidden convexity results open up the possibility to solve multi-stage robust optimization problems using certain nonlinear decision rules.
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Aharon Ben-Tal