Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
On approximation algorithms for concave quadratic programming
Recent advances in global optimization
The complexity of approximating a nonlinear program
Mathematical Programming: Series A and B
Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming
Journal of Global Optimization
LMI Approximations for Cones of Positive Semidefinite Forms
SIAM Journal on Optimization
FPTAS for mixed-integer polynomial optimization with a fixed number of variables
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A linear programming reformulation of the standard quadratic optimization problem
Journal of Global Optimization
Integer Polynomial Optimization in Fixed Dimension
Mathematics of Operations Research
The UGC hardness threshold of the ℓp Grothendieck problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Approximation Guarantees through Higher Levels of SDP Hierarchies
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
The UGC Hardness Threshold of the Lp Grothendieck Problem
Mathematics of Operations Research
SIAM Journal on Optimization
Maximum Block Improvement and Polynomial Optimization
SIAM Journal on Optimization
Approximation algorithm for a class of global optimization problems
Journal of Global Optimization
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We consider the problem of computing the minimum value pmin taken by a polynomial p(x) of degree d over the standard simplex Δ. This is an NP-hard problem already for degree d = 2. For any integer k ≥ 1, by minimizing p(x) over the set of rational points in Δ with denominator k, one obtains a hierarchy of upper bounds pΔ(k) converging to pmin as k → ∞. These upper approximations are intimately linked to a hierarchy of lower bounds for pmin constructed via Pólya's theorem about representations of positive forms on the simplex. Revisiting the proof of Pólya's theorem allows us to give estimates on the quality of these upper and lower approximations for pmin. Moreover, we show that the bounds pΔ(k) yield a polynomial time approximation scheme for the minimization of polynomials of fixed degree d on the simplex, extending an earlier result of Bomze and De Klerk for degree d = 2.