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A simple characterization of solutions sets of convex programs. (English) Zbl 0653.90055
The author shows that for any convex program $$\min_{x\in X}f(x)$$, where X is a convex set in R n and f(x) is a convex function on R n, the subdifferential $$\partial f(x)$$ is constant on the relative interior of the set $$\bar X=\arg \min_{x\in X}f(x)$$ (solution set of the problem) and equals the intersection of the subdifferentials of the function f(x) at all points of $$\bar X.$$ In addition, $$\bar X$$ lies in the intersection with the feasible set X of an affine subspace orthogonal to some subgradient of f(x) at a relative interior point of $$\bar X.$$ As a consequence a simple polyhedral characterization is given for the solution set of a convex quadratic program and that of a monotone linear complementarity problem.
Reviewer: H.Tuy

##### MSC:
 90C25 Convex programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C20 Quadratic programming
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##### References:
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