Related papers: Revisiting Degeneracy, Strict Feasibility, Stabili…
Slater's condition -- existence of a "strictly feasible solution" -- is a common assumption in conic optimization. Without strict feasibility, first-order optimality conditions may be meaningless, the dual problem may yield little…
This paper examines the feasible region of a standard conic program represented as the intersection of a closed convex cone and a set of linear equalities. It is recently shown that when Slater constraint qualification (strict feasibility)…
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear…
Strict linear feasibility or linear separation is usually tackled using efficient approximation/stochastic algorithms (that may even run in sub-linear times in expectation). However, today state of the art for solving…
Linear programming (LP) is an extremely useful tool and has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
The real radical ideal of a system of polynomials with finitely many complex roots is generated by a system of real polynomials having only real roots and free of multiplicities. It is a central object in computational real algebraic…
In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction,…
Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward…
We present a short step interior point method for solving a class of nonlinear programming problems with quadratic objective function. Convex quadratic programming problems can be reformulated as problems in this class. The method is shown…
A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified…
We show that the effects of finite-precision arithmetic in forming and solving the linear system that arises at each iteration of primal-dual interior-point algorithms for nonlinear programming are benign, provided that the iterates satisfy…
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.…
We review the simplex method and two interior-point methods (the affine scaling and the primal-dual) for solving linear programming problems for checking avoiding sure loss, and propose novel improvements. We exploit the structure of these…
Linear superiorization considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not…
We develop a new `subspace layered least squares' interior point method (IPM) for solving linear programs. Applied to an $n$-variable linear program in standard form, the iteration complexity of our IPM is up to an $O(n^{1.5} \log n)$…
A spectrahedron is the feasible set of a semidefinite program, SDP, i.e., the intersection of an affine set with the positive semidefinite cone. While strict feasibility is a generic property for random problems, there are many classes of…
Linear solvers are major computational bottlenecks in a wide range of decision support and optimization computations. The challenges become even more pronounced on heterogeneous hardware, where traditional sparse numerical linear algebra…
Facial reduction, FR, is a regularization technique for convex programs where the strict feasibility constraint qualification, CQ, fails. Though this CQ holds generically, failure is pervasive in applications such as semidefinite…
The linear programming (LP) approach is, together with value iteration and policy iteration, one of the three fundamental methods to solve optimal control problems in a dynamic programming setting. Despite its simple formulation,…