Related papers: The algebraic degree of sparse polynomial optimiza…

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…

The Zariski closure of the central path which interior point algorithms track in convex optimization problems such as linear, quadratic, and semidefinite programs is an algebraic curve. The degree of this curve has been studied in relation…

Polynomial optimization problems represent a wide class of optimization problems, with a large number of real-world applications. Current approaches for polynomial optimization, such as the sum of squares (SOS) method, rely on large-scale…

We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…

Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…

In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent…

We study the optimization problem over the weakly Pareto set of a convex multiobjective optimization problem given by polynomial functions. Using Lagrange multiplier expressions and the weight vector, we give three types of representations…

In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and…

In sparse optimization, enforcing hard constraints using the $\ell_0$ pseudo-norm offers advantages like controlled sparsity compared to convex relaxations. However, many real-world applications demand not only sparsity constraints but also…

Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…

Estimation of the precision matrix (or inverse covariance matrix) is of great importance in statistical data analysis and machine learning. However, as the number of parameters scales quadratically with the dimension $p$, computation…

This thesis focuses on Bayesian optimization with the improvements coming from two aspects:(i) the use of derivative information to accelerate the optimization convergence; and (ii) the consideration of scalable GPs for handling massive…

This paper examines a variety of classical optimization problems, including well-known minimization tasks and more general variational inequalities. We consider a stochastic formulation of these problems, and unlike most previous work, we…

We show that sparsity constrained optimization problems over low dimensional spaces tend to have a small duality gap. We use the Shapley-Folkman theorem to derive both data-driven bounds on the duality gap, and an efficient primalization…

We propose an operational degree of polarization in terms of the variance of the projected Stokes vector minimized over all the directions of the Poincar\'e sphere. We examine the properties of this degree and show that some problems…

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 study sets defined as the intersection of a rank-1 constraint with different choices of linear side constraints. We identify different conditions on the linear side constraints, under which the convex hull of the rank-1 set is polyhedral…

Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the $\ell_1$-norm. However, several important learning applications cannot benefit from this approach as…

Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gr\"obner bases in the 60s, there has been a lot of progress in this domain. Moreover, these…

We study computational and statistical consequences of problem geometry in stochastic and online optimization. By focusing on constraint set and gradient geometry, we characterize the problem families for which stochastic- and…