In this article we introduce an inner parallel cutting plane method (IPCP) to compute good feasible points along with valid cutting planes for mixed-integer convex optimization problems. The method iteratively generates polyhedral outer approximations of an enlarged inner parallel set (EIPS) of the continuously relaxed feasible set. This EIPS possesses the crucial property that any … Read more
We consider feasibility problems defined by a set of constraints that exhibit gradient H\”older continuity plus additional constraints defined by the affordability of obtaining approximate minimizers of quadratic models onto the associated feasible set. Each iteration of the method introduced in this paper involves the approximate minimization of a two-norm regularized quadratic subject to the … Read more
The Douglas-Rachford algorithm is a classical and a successful method for solving the feasibility problems. Here, we provide a region for global convergence of the algorithm for the feasibility problem involving a disc and a circle in the Euclidean space of dimension two. Citation 1. Borwein, J.M., Sims, B.: The Douglas-Rachford algorithm in the absence … Read more
We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual functions. For amenable cones, it is shown that error bounds can be expressed as a composition of facial residual functions. The … Read more
In this work we present an extension of Chubanov’s algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov’s method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and K. Following an … Read more
We suppose the existence of an oracle which solves any semidefinite programming (SDP) problem satisfying Slater’s condition simultaneously at its primal and dual sides. We note that such an oracle might not be able to directly solve general SDPs even after certain regularization schemes are applied. In this work we fill this gap and show … Read more
The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, … Read more
We present an algorithm for nonlinear least-squares and nonlinear feasibility problems, i.e. for systems of nonlinear equations and nonlinear inequalities, which depend on the outcome of expensive functions for which derivatives are assumed to be unavailable. Our algorithm combines derivative-free techniques with filter trust-region methods to keep the number of expensive function evaluations low and … Read more
The feasibility pump (FP) [5,7] has proved to be a successful heuristic for finding feasible solutions of mixed integer linear problems (MILPs). FP was improved in [1] for finding better quality solutions. Briefly, FP alternates between two sequences of points: one of feasible so- lutions for the relaxed problem (but not integer), and another of … Read more
We define a variant of Anstreicher and Terlaky’s (1994) monotonic build-up (MBU) simplex algorithm for linear feasibility problems. Under a nondegeneracy assumption weaker than the normal one, the complexity of the algorithm can be given by $m\Delta$, where $\Delta$ is a constant determined by the input data of the problem and $m$ is the number … Read more