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The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over the computationally intractable cone of completely positive matrices.…
Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program…
Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist…
This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems,…
We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite…
The success of several constraint-based modeling languages such as OPL, ZINC, or COMET, appeals for better software engineering practices, particularly in the testing phase. This paper introduces a testing framework enabling automated test…
A linear programming (LP) based framework is presented for obtaining converses for finite blocklength lossy joint source-channel coding problems. The framework applies for any loss criterion, generalizes certain previously known converses,…
We present exact mixed-integer linear programming formulations for verifying the performance of first-order methods for parametric quadratic optimization. We formulate the verification problem as a mixed-integer linear program where the…
Semidefinite programming (SDP) is a fundamental class of convex optimization problems with diverse applications in mathematics, engineering, machine learning, and related disciplines. This paper investigates the application of the…
Constrained partially observable Markov decision processes (CPOMDPs) have been used to model various real-world phenomena. However, they are notoriously difficult to solve to optimality, and there exist only a few approximation methods for…
This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes…
This paper introduces a computationally efficient method that converges globally to B-stationary points of mathematical programs with equilibrium constraints (MPECs). B-stationarity is necessary for optimality and means that no feasible…
We construct a general framework for deriving error bounds for conic feasibility problems. In particular, our approach allows one to work with cones that fail to be amenable or even to have computable projections, two previously challenging…
We discuss the application of random projections to conic programming: notably linear, second-order and semidefinite programs. We prove general approximation results on feasibility and optimality using the framework of formally real Jordan…
Sensitivity analysis plays a crucial role in multiobjective linear programming (MOLP), where understanding the impact of parameter changes on efficient solutions is essential. This work builds upon and extends previous investigations. In…
We introduce a verification framework to exactly verify the worst-case performance of sequential convex programming (SCP) algorithms for parametric non-convex optimization. The verification problem is formulated as an optimization problem…
The LP-Newton method solves the linear programming problem (LP) by repeatedly projecting a current point onto a certain relevant polytope. In this paper, we extend the algorithmic framework of the LP-Newton method to the second-order cone…
We investigate robust optimization problems defined for maximizing convex functions. For finite uncertainty set, we develop a geometric branch-and-bound algorithmic approach to solve this problem. The geometric branch-and-bound algorithm…
Based on smoothing techniques, we propose two new methods to solve linear complementarity problems (LCP) called TLCP and Soft-Max. The idea of these two new methods takes inspiration from interior-point methods in optimization. The…
Our contribution in this paper is two folded. We consider first the case of linear programming with real coefficients and give a method which allows the computation of a new upper bound on the distance from the origin to a feasible point.…