Papers by Tung Phan Thanh
49th IEEE Conference on Decision and Control (CDC), 2010
We present a new algorithm for solving a polynomial program P based on the recent "joint + margin... more We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for parametric polynomial optimization. The idea is to first consider the variable x1 as a parameter and solve the associated (n − 1)-variable (x2,. .. , xn) problem P(x1) where the parameter x1 is fixed and takes values in some interval Y1 ⊂ R, with some probability ϕ1 uniformly distributed on Y1. Then one considers the hierarchy of what we call "joint+marginal" semidefinite relaxations, whose duals provide a sequence of univariate polynomial approximations x1 → p k (x1) that converges to the optimal value function x1 → J(x1) of problem P(x1), as k increases. Then with k fixed a priori, one computesx * 1 ∈ Y1 which minimizes the univariate polynomial p k (x1) on the interval Y1, a convex optimization problem that can be solved via a single semidefinite program. The quality of the approximation depends on how large k can be chosen (in general for significant size problems k = 1 is the only choice). One iterates the procedure with now an (n − 2)variable problem P(x2) with parameter x2 in some new interval Y2 ⊂ R, etc. so as to finally obtain a vectorx ∈ R n. Preliminary numerical results are provided.
Journal of Global Optimization, 2012
We present a new algorithm for solving a polynomial program P based on the recent "joint + margin... more We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for parametric polynomial optimization. The idea is to first consider the variable x1 as a parameter and solve the associated (n − 1)-variable (x2,. .. , xn) problem P(x1) where the parameter x1 is fixed and takes values in some interval Y1 ⊂ R, with some probability ϕ1 uniformly distributed on Y1. Then one considers the hierarchy of what we call "joint+marginal" semidefinite relaxations, whose duals provide a sequence of univariate polynomial approximations x1 → p k (x1) that converges to the optimal value function x1 → J(x1) of problem P(x1), as k increases. Then with k fixed a priori, one computesx * 1 ∈ Y1 which minimizes the univariate polynomial p k (x1) on the interval Y1, a convex optimization problem that can be solved via a single semidefinite program. The quality of the approximation depends on how large k can be chosen (in general for significant size problems k = 1 is the only choice). One iterates the procedure with now an (n − 2)variable problem P(x2) with parameter x2 in some new interval Y2 ⊂ R, etc. so as to finally obtain a vectorx ∈ R n. Preliminary numerical results are provided.
Journal of Global Optimization, 2012
Convex underestimators of a polynomial on a box. Given a non convex polynomial f ∈ R[x] and a box... more Convex underestimators of a polynomial on a box. Given a non convex polynomial f ∈ R[x] and a box B ⊂ R n , we construct a sequence of convex polynomials (f dk) ⊂ R[x], which converges in a strong sense to the "best" (convex and degree-d) polynomial underestimator f * d of f. Indeed, f * d minimizes the L1-norm f − g 1 on B, over all convex degree-d polynomial underestimators g of f. On a sample of problems with non convex f , we then compare the lower bounds obtained by minimizing the convex underestimator of f computed as above and computed via the popular αBB method. In all examples we obtain significantly better results.
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Papers by Tung Phan Thanh