Related papers: Probabilistic Algorithm for Polynomial Optimizatio…
We propose a novel distribution-free scheme to solve optimization problems where the goal is to minimize the expected value of a cost function subject to probabilistic constraints. Unlike standard sampling-based methods, our idea consists…
We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm BISOLVE. BISOLVE is a classical elimination method which first projects the solutions of a system onto the $x$-…
Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. This paper studies ProSparse, a new polynomial complexity algorithm that solves the sparse representation…
We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…
Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolic algorithm to find the equations and inequalities defining a semi-algebraic set $Q$ which is identical to the closure of the image of $S$ under $f$, i.e.,…
An algorithm is proposed, analyzed, and tested experimentally for solving stochastic optimization problems in which the decision variables are constrained to satisfy equations defined by deterministic, smooth, and nonlinear functions. It is…
The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes…
This article considers stochastic algorithms for efficiently solving a class of large scale non-linear least squares (NLS) problems which frequently arise in applications. We propose eight variants of a practical randomized algorithm where…
We design and analyze an algorithm for first-order stochastic optimization of a large class of functions on $\mathbb{R}^d$. In particular, we consider the \emph{variationally coherent} functions which can be convex or non-convex. The…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
We consider a class of stochastic programs whose uncertain data has an exponential number of possible outcomes, where scenarios are affinely parametrized by the vertices of a tractable binary polytope. Under these conditions, we propose a…
We consider the problem of computing a positive definite $p \times p$ inverse covariance matrix aka precision matrix $\theta=(\theta_{ij})$ which optimizes a regularized Gaussian maximum likelihood problem, with the elastic-net regularizer…
Contemporary global optimization algorithms are based on local measures of utility, rather than a probability measure over location and value of the optimum. They thus attempt to collect low function values, not to learn about the optimum.…
We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its…
We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…
In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. By using a technique in computer algebra and the second-order optimality condition, we provide a…
Optimization is a key task in a number of applications. When the set of feasible solutions under consideration is of combinatorial nature and described in an implicit way as a set of constraints, optimization is typically NP-hard.…
The main purpose of this paper is to study the NP-complete subset-sum problem, not in the usual context of time-complexity-based classification of the algorithms (exponential/polynomial), but through a new kind of algorithmic classification…