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The method of periodic projections consists in iterating projections onto $m$ closed convex subsets of a Hilbert space according to a periodic sweeping strategy. In the presence of $m\geq 3$ sets, a long-standing question going back to the…
One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well-known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact…
We consider projection algorithms for solving (nonconvex) feasibility problems in Euclidean spaces. Of special interest are the Method of Alternating Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection Algorithm…
Writing an uncomplicated, robust, and scalable three-dimensional convex hull algorithm is challenging and problematic. This includes, coplanar and collinear issues, numerical accuracy, performance, and complexity trade-offs. While there are…
This paper considers a networked system with a finite number of users and supposes that each user tries to minimize its own private objective function over its own private constraint set. It is assumed that each user's constraint set can be…
In this paper, we study randomized and cyclic coordinate descent for convex unconstrained optimization problems. We improve the known convergence rates in some cases by using the numerical semidefinite programming performance estimation…
We address the problem of projecting a point onto a quadratic hypersurface, more specifically a central quadric. We show how this problem reduces to finding a given root of a scalar-valued nonlinear function. We completely characterize one…
We propose a new geometric method of IR factorization in sector decomposition. The problem is converted into a set of problems in convex geometry. The latter problems are solved using algorithms in combinatorial geometry. This method…
This article focuses on numerical efficiency of projection algorithms for solving linear optimization problems. The theoretical foundation for this approach is provided by the basic result that bounded finite dimensional linear optimization…
We present a real-time algorithm that finds the Penetration Depth (PD) between general polygonal models based on iterative and local optimization techniques. Given an in-collision configuration of an object in configuration space, we find…
We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivator behind this numerical method is estimating solutions to Adverse Selection…
The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range…
Convexity, though extremely important in mathematical programming, has not drawn enough attention in the field of dynamic programming. This paper gives conditions for verifying convexity of the cost-to-go functions, and introduces an…
The article is devoted to the development of algorithmic methods ensuring efficient complexity bounds for strongly convex-concave saddle point problems in the case when one of the groups of variables is high-dimensional, and the other is…
Mixed-integer quadratic programs (MIQPs) are a versatile way of formulating vehicle decision making and motion planning problems, where the prediction model is a hybrid dynamical system that involves both discrete and continuous decision…
The concept of a visible point of a convex set relative to a given point is introduced. A number of basic properties of such visible point sets is developed. In particular, it is shown that this concept is useful in the study of best…
In this work we study the method of Bregman projections for deterministic and stochastic convex feasibility problems with three types of control sequences for the selection of sets during the algorithmic procedure: greedy, random, and…
In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately…
We study the integrality gap of convex mixed-integer programs, that is, the difference between the optimal value of such a problem and the optimal value of its continuous relaxation. We study classes of convex sets whose associated…
This article introduces a numerical algorithm that serves as a preliminary step toward solving continuous-time model predictive control (MPC) problems directly without explicit time-discretization. The chief ingredients of the underlying…