Convergence behavior of decomposition algorithms for linear programs

This paper discusses plausible explanations of the somewhat folkloric, ‘tailing off’ convergence behavior of the Dantzig-Wolfe decomposition algorithm for linear programs. Is is argued that such beahvior may be used to numerical inaccuracy. Procedures to identify and mitigate such difficulties are outlined.     

Convergence rates of cascade algorithms

We consider solutions of a refinement equation of the form

where is a finitely supported sequence called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the cascade algorithm associated with , i.e., the convergence of the sequence in the -norm.

Our main result gives estimates for the convergence rate of the cascade algorithm. Let be the normalized solution of the above refinement equation with the dilation matrix being isotropic. Suppose lies in the Lipschitz space , where 0$"> and . Under appropriate conditions on , the following estimate will be established:

where and is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.

     

Convergence of interval-type algorithms for generalized fractional programming

The purpose of this paper is to analyze the convergence of interval-type algorithms for solving the generalized fractional program. They are characterized by an interval [LB _k , UB _k ] including*, and the length of the interval is reduced at each iteration. A closer analysis of the bounds LB _k and UB _k allows to modify slightly the best known interval-type algorithm NEWMODM accordingly to prove its convergence and derive convergence rates similar to those for a Dinkelbach-type algorithm MAXMODM under the same conditions. Numerical results in the linear case indicate that the modifications to get convergence results are not obtained at the expense of the numerical efficiency since the modified version BFII is as efficient as NEWMODM and more efficient than MAXMODM.This research was supported by NSERC (Grant A8312) and FCAR (Grant 0899).     

Convergence of Implementable Descent Algorithms for Unconstrained Optimization

Descent algorithms use sufficient descent directions combined with stepsize rules, such as the Armijo rule, to produce sequences of iterates whose cluster points satisfy some necessary optimality conditions. In this note, we present a proof that the whole sequence of iterates converges for quasiconvex objective functions.     

Convergence theory of exact interpolation scheme for computing several eigenvectors

An asymptotic convergence analysis of a new multilevel method for numerical solution of eigenvalues and eigenvectors of symmetric and positive definite matrices is performed. The analyzed method is a generalization of the original method that has recently been proposed by R. Ku?el and P. Vaněk (DOI: 10.1002/nla.1975) and uses a standard multigrid prolongator matrix enriched by one full column vector, which approximates the first eigenvector. The new generalized eigensolver is designed to compute eigenvectors. Their asymptotic convergence in terms of the generalized residuals is proved, and its convergence factor is estimated. The theoretical analysis is illustrated by numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Convergence proof of minimization algorithms for nonconvex functions

In this paper, we prove a theorem of convergence to a point for descent minimization methods. When the objective function is differentiable, the convergence point is a stationary point. The theorem, however, is applicable also to nondifferentiable functions. This theorem is then applied to prove convergence of some nongradient algorithms.     

The convergence of broyden algorithms for LC gradient function

1. IntroductionWe know that the variable metric algorithms, such as the Broyden algorithms, are veryefficient methods for solving the nonlinear programming problem:With exact line search, [11 proved that all the Broyden algorithms produce the same iterations for general functions. [21 proved that the rate of convergellce of these algorithms isone-step superlinear for the uniformly convex object function, and [3] proved that if thepoints given by these algorithms are convergent they are globall…     

Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming

We prove the superlinear convergence of the primal-dual infeasible interior-point path-following algorithm proposed recently by Kojima, Shida, and Shindoh and by the present authors, under two conditions: (i) the semidefinite programming problem has a strictly complementary solution; (ii) the size of the central path neighborhood approaches zero. The nondegeneracy condition suggested by Kojima, Shida, and Shindoh is not used in our analysis. Our result implies that the modified algorithm of Kojima, Shida, and Shindoh, which enforces condition (ii) by using additional corrector steps, has superlinear convergence under the standard assumption of strict complementarity. Finally, we point out that condition (ii) can be made weaker and show the superlinear convergence under the strict complementarity assumption and a weaker condition than (ii).     

Global convergence rates for descent algorithms: Improved results for strictly convex problems

Convergence to the minimal value is studied for the important type of descent algorithm which, at each interation, uses a search direction making an angle with the negative gradient which is smaller than a prespecified angle. Improvements on existing convergence rate results are obtained.Paper received on 4 October, 1977; in revised form, April 3, 1978     

Convergence of cascade algorithms associated with nonhomogeneous refinement equations

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form where is the unknown, is a given vector of functions on , is an dilation matrix, and is a finitely supported refinement mask such that each is an (complex) matrix. Let be an initial vector in . The corresponding cascade algorithm is given by In this paper we give a complete characterization for the -convergence of the cascade algorithm in terms of the refinement mask , the nonhomogeneous term , and the initial vector of functions .

     

Analysis of decomposition algorithms via nonlinear splitting functions

This paper provides a structural analysis of decomposition algorithms using a generalization of linear splitting methods. This technique is used to identify explicitly the essential similarities and differences between several classical algorithms. Similar concepts can be used to analyze a large class of multilevel hierarchical structures.This research was supported in part by ONR Contract No. N00014-76-C-0346, in part by the US Department of Energy, Division of Electric Energy Systems, Contract No. ERDA-E(49-18)-2087 at the Massachusetts Institute of Technology, and in part by the Joint Services Electronics Program, Contract No. DAAG-29-78-C-0016.The authors would like to thank Dr. P. Varaiya, University of California at Berkeley, and Dr. D. Bertsekas for their comments and suggestions.     

Measuring rates of convergence of numerical algorithms

We analyze the behavior of common indices used in numerical linear algebra, analysis, and optimization to measure rates of convergence of an algorithm. A simple consistent axiomatic structure is used to uniquely define convergence rate measures on the basic linear, superlinear, and sublinear scales in terms of standard comparison sequences. Agreement with previously utilized indices and related measures is discussed.This research was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada.The authors are grateful to the referees for comments which improved an earlier draft.     

Augmented Lagrangian algorithms for linear programming

We introduce new augmented Lagrangian algorithms for linear programming which provide faster global convergence rates than the augmented algorithm of Polyak and Treti'akov. Our algorithm shares the same properties as the Polyak-Treti'akov algorithm in that it terminates in finitely many iterations and obtains both primal and dual optimal solutions. We present an implementable version of the algorithm which requires only approximate minimization at each iteration. We provide a global convergence rate for this version of the algorithm and show that the primal and dual points generated by the algorithm converge to the primal and dual optimal set, respectively.     

Local search algorithms for finding the Hamiltonian completion number of line graphs

Given a graph G=(V,E), the Hamiltonian completion number of G, HCN(G), is the minimum number of edges to be added to G to make it Hamiltonian. This problem is known to be -hard even when G is a line graph. In this paper, local search algorithms for finding HCN(G) when G is a line graph are proposed. The adopted approach is mainly based on finding a set of edge-disjoint trails that dominates all the edges of the root graph of G. Extensive computational experiments conducted on a wide set of instances allow to point out the behavior of the proposed algorithms with respect to both the quality of the solutions and the computation time.     

Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms

The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization problems (Refs. 1–3) and variational inequalities (Ref. 4). The key assumption to prove the global and strong convergence of such algorithms, as well as of most of the other algorithms proposed in the literature, is the strong monotony of the operator involved in the variational inequalities. In this paper, we consider variational inequalities defined over a product of spaces and we introduce a new property of strong nested monotony, which is weaker than the ordinary overall strong monotony generally assumed. In some sense, this new concept seems to be a minimal requirement to insure convergence of the algorithms alluded to above. A convergence theorem based on this weaker assumption is given. Application of this result to the computation of Nash equilibria can be found in another paper (Ref. 5).This research has been supported by the Centre National de la Recherche Scientifique (ATP Complex Technological Systems) and by the Centre National d'Etudes des Télécommunications (Contract 83.5B.034.PAA).     

Global Convergence of Conjugate Gradient Methods without Line Search

Global convergence results are derived for well-known conjugate gradient methods in which the line search step is replaced by a step whose length is determined by a formula. The results include the following cases: (1) The Fletcher–Reeves method, the Hestenes–Stiefel method, and the Dai–Yuan method applied to a strongly convex LC ¹ objective function; (2) The Polak–Ribière method and the Conjugate Descent method applied to a general, not necessarily convex, LC ¹ objective function.     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.     

Convergence qualification of adaptive partition algorithms in global optimization

Following the presentation of a general partition algorithm scheme for seeking the globally best solution in multiextremal optimization problems, necessary and sufficient convergence conditions are formulated, in terms of respectively implied or postulated properties of the partition operator. The convergence results obtained are pertinent to a number of deterministic algorithms in global optimization, permitting their diverse modifications and generalizations.