Generating feasible points for mixed-integer convex optimization problems by inner parallel cuts

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

On the complexity of solving feasibility problems

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

Douglas-Rachford method for the feasibility problem involving a circle and a disc

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

Amenable cones: error bounds without constraint qualifications

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

An extension of Chubanov’s algorithm to symmetric cones

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

Solving SDP Completely with an Interior Point Oracle

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

Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

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

Solving structured nonlinear least-squares and nonlinear feasibility problems with expensive functions

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

Using the analytic center in the feasibility pump

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

Anstreicher-Terlaky type monotonic simplex algorithms for linear feasibility problems

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