Related papers: A "joint+marginal" approach to parametric polynomi…
A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial…
We present a general technique, based on parametric search with some twist, for solving a variety of optimization problems on a set of semi-algebraic geometric objects of constant complexity. The common feature of these problems is that…
We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time $(0,\epsilon)$-additive bicriteria approximation algorithm for finding optimal…
The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex…
We propose a black-box approach to reducing large semidefinite programs to a set of smaller semidefinite programs by projecting to random linear subspaces. We evaluate our method on a set of polynomial optimization problems, demonstrating…
We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gr\"obner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation…
We consider the fundamental learning problem of estimating properties of distributions over large domains. Using a novel piecewise-polynomial approximation technique, we derive the first unified methodology for constructing sample- and…
When model predictions inform downstream decision making, a natural question is under what conditions can the decision-makers simply respond to the predictions as if they were the true outcomes. Calibration suffices to guarantee that simple…
Big data is ubiquitous in practices, and it has also led to heavy computation burden. To reduce the calculation cost and ensure the effectiveness of parameter estimators, an optimal subset sampling method is proposed to estimate the…
We present a technique for entropy optimization to calculate a distribution from its moments. The technique is based upon maximizing a discretized form of the Shannon entropy functional by mapping the problem onto a dual space where an…
We consider a discrete best approximation problem formulated in the framework of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. Given a set of samples of input and output of…
We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof…
We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of…
We consider polynomial optimization problems on Cartesian products of basic compact semialgebraic sets. The solution of such problems can be approximated as closely as desired by hierarchies of semidefinite programming relaxations, based on…
We consider the problem of constructing an approximation of the Pareto curve associated with the multiobjective optimization problem $\min_{\mathbf{x} \in \mathbf{S}}\{ (f_1(\mathbf{x}), f_2(\mathbf{x})) \}$, where $f_1$ and $f_2$ are two…
We consider the problem of finding the global optimum of a real-valued complex polynomial on a compact set defined by real-valued complex polynomial inequalities. It reduces to solving a sequence of complex semidefinite programming…
This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to…
Deciding what to sense is a crucial task, made harder by dependencies and by a nonadditive utility function. We develop approximation algorithms for selecting an optimal set of measurements, under a dependency structure modeled by a…
Parametric optimization solves a family of optimization problems as a function of parameters. It is a critical component in situations where optimal decision making is repeatedly performed for updated parameter values, but computation…
We study the polynomial optimization problem of minimizing a multihomogeneous polynomial over the product of spheres. This polynomial optimization problem models the tensor optimization problem of finding the best rank one approximation of…