Journal of Symbolic Computation
Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems
Numerische Mathematik
Numerical analysis: an introduction
Numerical analysis: an introduction
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Approximation Schemes for Infinite Linear Programs
SIAM Journal on Optimization
Polynomial Programming: LP-Relaxations Also Converge
SIAM Journal on Optimization
Optimization of Polynomials on Compact Semialgebraic Sets
SIAM Journal on Optimization
SIAM Journal on Optimization
Persistence in discrete optimization under data uncertainty
Mathematical Programming: Series A and B
Convergent SDP-Relaxations in Polynomial Optimization with Sparsity
SIAM Journal on Optimization
Persistency Model and Its Applications in Choice Modeling
Management Science
GloptiPoly 3: moments, optimization and semidefinite programming
Optimization Methods & Software - GLOBAL OPTIMIZATION
Hi-index | 0.00 |
Given a compact parameter set $\mathbf{Y}\subset\mathbb{R}^p$, we consider polynomial optimization problems $(\mathbf{P}_{\mathbf{y}}$) on $\mathbb{R}^n$ whose description depends on the parameter $\mathbf{y}\in\mathbf{Y}$. We assume that one can compute all moments of some probability measure $\varphi$ on $\mathbf{Y}$, absolutely continuous with respect to the Lebesgue measure (e.g., $\mathbf{Y}$ is a box or a simplex and $\varphi$ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions $\mathbf{x}^*(\mathbf{y})$ of $\mathbf{P}_{\mathbf{y}}$, as $\mathbf{y}\in\mathbf{Y}$. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions like, e.g., their $\varphi$-mean. In addition, using this knowledge on moments, the measurable function $\mathbf{y}\mapsto x^*_k(\mathbf{y})$ of the $k$th coordinate of optimal solutions, can be estimated, e.g., by maximum entropy methods. Also, for a boolean variable $x_k$, one may approximate as closely as desired its persistency $\varphi(\{\mathbf{y}:x^*_k(\mathbf{y})=1\}$, i.e., the probability that in an optimal solution $\mathbf{x}^*(\mathbf{y})$, the coordinate $x^*_k(\mathbf{y})$ takes the value 1. Last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp., piecewise polynomial) lower approximations with $L_1(\varphi)$ (resp., $\varphi$-almost uniform) convergence to the optimal value function.