Related papers: Higher Order Maximum Persistency and Comparison Th…
Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of…
In this paper, we investigate several properties of the solution maps of variational inequalities with polynomial data. First, we prove some facts on the $R_0$-property, the local boundedness, and the upper semicontinuity of the solution…
We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lov\'{a}sz--Schrijver lift-and-project hierarchy. Our…
The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of…
In this work, we reveal a rich combinatorial structure underlying exact minimax optimal algorithms for classical nonexpansive fixed-point problems. This viewpoint unifies all extremal optimal methods and provides a systematic and practical…
We consider the problem of extracting a low-dimensional, linear latent variable structure from high-dimensional random variables. Specifically, we show that under mild conditions and when this structure manifests itself as a linear space…
In recent years, bilevel approaches have become very popular to efficiently estimate high-dimensional hyperparameters of machine learning models. However, to date, binary parameters are handled by continuous relaxation and rounding…
We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use…
We consider continuous linear programs over a continuous finite time horizon $T$, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions, where we search for optimal solutions in the space…
Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes…
We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving…
We study the fixed point problem for a system of multivariate operators that are coordinate-wise monotone (i.e., nondecreasing or nonincreasing in each of the variables, independently), in the setting of quasi-ordered sets. We show that…
We study the problem of finding solutions to the stable matching problem that are robust to errors in the input and we obtain a polynomial time algorithm for a special class of errors. In the process, we also initiate work on a new…
This paper addresses a class of problems under interval data uncertainty composed of min-max regret versions of classical 0-1 optimization problems with interval costs. We refer to them as interval 0-1 min-max regret problems. The…
In this paper we assemble some results about the upper-semicontinuity and lower-semicontinuity of the feasible correspondence and the solution correspondence of linear programming problems allowing variability of all parameters of such…
Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting…
In multi-objective optimization, a single decision vector must balance the trade-offs between many objectives. Solutions achieving an optimal trade-off are said to be Pareto optimal: these are decision vectors for which improving any one…