On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems
Etienne De Klerk (e.deklerkuvt.nl)
Abstract: The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995-2014, 2009.] We give a new proof of the finite convergence property, that does not require the assumption that the Hessian of the objective be positive definite on the entire feasible set, but only at the optimal solution. In addition, we show that the number of steps needed for convergence depends on more than the input size of the problem. In particular, the size of the semidefinite program that gives the exact reformulation of the convex polynomial optimization problem may be exponential in the input size.
Keywords: convex polynomial optimization, sum of squares of polynomials, positivstellensatz, semidefinite programming
Category 1: Convex and Nonsmooth Optimization
Category 2: Linear, Cone and Semidefinite Programming
Citation: Manuscript, Tilburg University, The Netherlands, November 2010 SIAM J. Optimization, to appear.
Entry Submitted: 11/07/2010
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