Convex Feasibility Problem Research Articles

A Polynomial Cutting Surfaces Algorithm for the ConvexFeasibility Problem Defined by Self-Concordant Inequalities

Consider a nonempty convex set in R^m which is defined by a finite number of smooth convex inequalities and which admits a self-concordant logarithmic barrier. We study the analytic center based column generation algorithm for the problem of finding a feasible point in this set. At each iteration the algorithm computes an approximate analytic center of the set defined by the inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminatess otherwise either an existing inequality is shifted or a new inequality is added into the system. As the number of iterations increases, the set defined by the generated inequalities shrinks and the algorithm eventually finds a solution of the problem. The algorithm can be thought of as an extension of the classical cutting plane method. The difference is that we use analytic centers and “convex cuts” instead of arbitrary infeasible points and linear cuts. In contrast to the cutting plane method, the algorithm has a polynomial worst case complexity ofO(N\log\frac{1}{\varepsilon}) on the total number of cuts to be used, where N is the number of convex inequalities in the original problem and e is the maximum common slack of the original inequality system.

Iterative projection onto convex sets using multiple Bregman distances

A number of inverse problems, in image reconstruction and elsewhere, can be formulated in terms of finding a vector in the intersection of certain convex sets that serve to constrain the solution. Finding such vectors is called the `convex feasibility problem' (CFP). Algorithms to solve the CFP are usually iterative `projection onto convex sets' (POCS) methods that employ orthogonal or more general projections onto the individual convex sets; Bregman's `successive generalized projection' (SGP) is one such method. When the intersection of the convex sets is heterogeneous one may wish to optimize a certain function over that intersection; then we have a constrained optimization problem. Generalized projections come from generalized distances, typically Bregman distances, chosen to incorporate prior information, such as non-negativity, about the image being reconstructed. Calculating a generalized projection onto a convex set can be simplified if the generalized distance can vary with the convex set. Censor and Elfving have discovered a simultaneous multiprojection algorithm that permits just such variation. Because simultaneous methods, which use all the convex sets at each step, can be slow to converge, there is considerable interest in faster, block-iterative methods that employ only some of the convex sets at each step. In this paper we present the first such method that permits multiple generalized projections. We introduce a new notion of convex combination and apply it to obtain an extension of the SGP, called the `multidistance SGP' (MSGP) method, that allows for projecton with respect to multiple generalized distances. We conclude with an extension of the MSGP to block-iterative algorithms involving relaxed generalized projections and paracontractive operators.

Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization

The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson's duality for two cones, which relates bounded linear regularity to property (G), is re-derived and refined. For polyhedral cones, a statement dual to Hoffman's error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces.

Primal–Dual–Infeasible Newton Approach for the Analytic Center Deep-Cutting Plane Method

The convergence and complexity of a primal–dual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a deep cut separation oracle is studied. The primal–dual–infeasible Newton method is used to generate a primal–dual updating direction. The number of recentering steps is O(1) for cuts as deep as half way to the deepest cut, where the deepest cut is tangent to the primal–dual variant of Dikin's ellipsoid.

Homogeneous analytic center cutting plane methods with approximate centers

In this paper we consider a homogeneous analytic center cutting plane method in a projective space. We describe a general scheme that uses a homogeneous oracle and computes an approximate analytic center at each iteration. This technique is applied to a convex feasibility problem, to variational inequalities, and to convex constrained minimization. We prove that these problems can be solved with the same order of complexity as in the case of exact analytic centers. For the feasibility and the minimization problems rough approximations suffice, but very high precision is required for the variational inequalities. We give an example of variational inequality where even the first analytic center needs to be computed with a precision matching the precision required for the solution.

Generalized Bregman Projections in Convex Feasibility Problems

We present a method for finding common points of finitely many closed convex sets in Euclidean space. The Bregman extension of the classical method of cyclic orthogonal projections employs nonorthogonal projections induced by a convex Bregman function, whereas the Bauschke and Borwein method uses Bregman/Legendre functions. Our method works with generalized Bregman functions (B-functions) and inexact projections, which are easier to compute than the exact ones employed in other methods. We also discuss subgradient algorithms with Bregman projections.

An Analytic Center Based Column Generation Algorithm for Convex Quadratic Feasibility Problems

We consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities. At each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut (defined by a convex quadratic inequality) is placednear the approximate center. As the number of cuts increases, the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem. Previously, similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem. Our method differs from these early methods in that we use "quadratic cuts" in the computation instead of linear cuts. Moreover, our method has a polynomial worst case complexity of $O(n\ln \frac{1}{\varepsilon})$ on the total number of cuts to be used, where n is the number of convex quadratic polynomial inequalities in the problem and $\varepsilon$ is the radius of the largest ball contained in the feasible set. In contrast, the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time.

Incomplete projection algorithms for solving the convex feasibility problem

We present a general scheme for solving the convex feasibility problem and prove its convergence under mild conditions. Unlike previous schemes no exact projections are required. Moreover, we also introduce an acceleration factor, which we denote as the λ factor, that seems to play a fundamental role to improve the quality of convergence. Numerical tests on systems of linear inequalities randomly generated give impressive results in a multi-processing environment. The speedup is superlinear in some cases. New acceleration techniques are proposed, but no tests are reported here. As a by-product we obtain the rather surprising result that the relaxation factor, usually confined to the interval (0,2), gives better convergence results for values outside this interval.

Surrogate Projection Methods for Finding Fixed Points of Firmly Nonexpansive Mappings

We present methods for finding common fixed points of finitely many firmly nonexpansive mappings on a Hilbert space. At every iteration, an approximation to each mapping generates a halfspace containing its set of fixed points. The next iterate is found by projecting the current iterate on a surrogate halfspace formed by taking a convex combination of the halfspace inequalities. This acceleration technique extends one for convex feasibility problems (CFPs), since projection operators onto closed convex sets are firmly nonexpansive. The resulting methods are block iterative and, hence, lend themselves to parallel implementation. We extend to accelerated methods some recent results of Bauschke and Borwein [SIAM Rev., 38 (1996), pp. 367--426] on the convergence of projection methods.

A Scaled L 2 Optimization Approach for Improving Sensitivity of H ∞ Detection Filters for LTV Systems

Abstract In this paper the robust detection filter design problem as a scaled ο filtering problem is presented where the effect of estimation weighting on filtering performance is optimized through input diagonal scaling of the estimation. The objective is that, by the proper choice of the estimation weight, provide the smallest scaled L gain of the disturbance input of the system that is guaranteed to be less than a prespecified positive constant i.e., to derive the filter with optimal disturbance suppression capability. It is shown how to obtain bounds on the scaled L gain of the system by transforming the standard ο filtering problem into a convex feasibility problem. Numerical results demonstrating the effect of the scaled optimization are presented.

Hilbertian convex feasibility problem: Convergence of projection methods

The classical problem of finding a point in the intersection of countably many closed and convex sets in a Hilbert space is considered. Extrapolated iterations of convex combinations of approximate projections onto subfamilies of sets are investigated to solve this problem. General hypotheses are made on the regularity of the sets and various strategies are considered to control the order in which the sets are selected. Weak and strong convergence results are established within thisbroad framework, which provides a unified view of projection methods for solving hilbertian convex feasibility problems.

Solving convex feasibility problems by a parallel projection method with geometrically-defined parameters

We present a procedure for solving convex feasibility problems by a paralel projection method in which at each interation step the variable weights and the variable relaxation coefficient are determined by geometric criteria.

Monotone gram matrices and deepest surrogate inequalities in accelerated relaxation methods for convex feasibility problems

The relaxation method for linear inequalities iterates by projecting the current point onto the most violated constraint. Accelerated methods project onto the intersection of several halfspaces or onto a surrogate halfspace corresponding to a nonnegative combination of constraints. We extend Todd's conditions for finding best surrogate inequalities via the solution of systems of linear equations. Our techniques may be used for accelerating various methods for convex feasibility and optimization problems.

Robustness analysis and synthesis for nonlinear uncertain systems

A state-space characterization of robustness analysis and synthesis for nonlinear uncertain systems is proposed. The robustness of a class of nonlinear systems subject to L/sub 2/-bounded structured uncertainties is characterized in terms of a nonlinear matrix inequality (NLMI), which yields a convex feasibility problem. As in the linear case, scalings are used to find a Lyapunov or storage function that give sufficient conditions for robust stability and performances. Sufficient conditions for the solvability of robustness synthesis problems are represented in terms of NLMIs as well. With the proposed NLMI characterizations, it is shown that the computation needed for robustness analysis and synthesis is not more difficult than that for checking Lyapunov stability; the numerical solutions for robustness problems are approximated by the use of finite element methods or finite difference schemes, and the computations are reduced to solving linear matrix inequalities. Unfortunately, while the development in this paper parallels the corresponding linear theory, the resulting computational consequences are, of course, not as favourable.

Iterative Averaging of Entropic Projections for Solving Stochastic Convex Feasibility Problems

The problem considered in this paper is that of finding a point which is common to almost all the members of a measurable family of closed convex subsets of {R}_{++}^n, provided that such a point exists. The main results show that this problem can be solved by an iterative method essentially based on averaging at each step the Bregman projections with respect to f(x)=σ_i=1^nx_i\cdot \ln x_i of the current iterate onto the given sets.

Convergence of Bregman Projection Methods for Solving Consistent Convex Feasibility Problems in Reflexive Banach Spaces

The problem that we consider is whether or under what conditions sequences generated in reflexive Banach spaces by cyclic Bregman projections on finitely many closed convex subsets Q i with nonempty intersection converge to common points of the given sets.

An Adaptive Parallel Projection Method for Solving Convex Feasibility Problems

An Adaptive Parallel Projection Method for Solving Convex Feasibility Problems

On Projection Algorithms for Solving Convex Feasibility Problems

Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.

Complexity Analysis of an Interior Cutting Plane Method for Convex Feasibility Problems

We further analyze the convergence and the complexity of a dual column generation algorithm for solving general convex feasibility problems defined by a separation oracle. The oracle is called at an approximate analytic center of the set given by the intersection of the linear inequalities which are the previous answers of the oracle. We show that the algorithm converges in finite time and is in fact a fully polynomial approximation algorithm, provided that the feasible region has a nonempty interior.

The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Space

Determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences. An algorithm for finding common fixed points of nonexpansive mappings in Hilbert space, essentially due to Halpern, is analyzed. The main theorem extends Wittmann's recent work and partially generalizes a result by Lions. Algorithms of this kind have been applied to the convex feasibility problem.