An approximate subgradient algorithm for unconstrained nonsmooth,nonconvex optimization

In this paper a new algorithm for minimizing locally Lipschitz functions is developed. Descent directions in this algorithm are computed by solving a system of linear inequalities. The convergence of the algorithm is proved for quasidifferentiable semismooth functions. We present the results of numerical experiments with both regular and nonregular objective functions. We also compare the proposed algorithm with two different versions of the subgradient method using the results of numerical experiments. These results demonstrate the superiority of the proposed algorithm over the subgradient method.      

Penalty Function Approach to Linear Trilevel Programming

In this paper, we study a trilevel decision-making situation in which decisionmaker 1 selects an action, within a specified constraint set, then decisionmaker 2 selects an action within a constraint set determined by the action of decisionmaker 2, and finally decisionmaker 3 selects an action within a constraint set determined by the actions of decisionmakers 1 and 2. Each decisionmaker has an objective function to be optimized within the imposed constraint set. Bard (Ref. 1) presents a five-step algorithm for solving this problem. This paper presents an alternative approach to the key first step of that algorithm, which has some qualitative advantages over it.     

Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems

In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented. Supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

Algorithmic Issues for Electromagnetic Scattering in Layered Media: Green's Functions, Current Basis, and Fast Solver

In this paper, we will address three computational issues for electromagnetic scattering in layered media: fast algorithm of calculating dyadic Green's functions, high order RWG current basis and fast solver for layered media. Numerical results will be included.     

A recursive quadratic programming algorithm that uses differentiable exact penalty functions

In this paper, a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied. The line search functions used are approximations to Fletcher's differentiable exact penalty function. Global convergence and local superlinear convergence results are proved, and some numerical results are given.     

A new modularity measure for Fuzzy Community detection problems based on overlap and grouping functions

One of the main challenges of fuzzy community detection problems is to be able to measure the quality of a fuzzy partition. In this paper, we present an alternative way of measuring the quality of a fuzzy community detection output based on n-dimensional grouping and overlap functions. Moreover, the proposed modularity measure generalizes the classical Girvan–Newman (GN) modularity for crisp community detection problems and also for crisp overlapping community detection problems. Therefore, it can be used to compare partitions of different nature (i.e. those composed of classical, overlapping and fuzzy communities). Particularly, as is usually done with the GN modularity, the proposed measure may be used to identify the optimal number of communities to be obtained by any network clustering algorithm in a given network. We illustrate this usage by adapting in this way a well-known algorithm for fuzzy community detection problems, extending it to also deal with overlapping community detection problems and produce a ranking of the overlapping nodes. Some computational experiments show the feasibility of the proposed approach to modularity measures through n-dimensional overlap and grouping functions.     

Univariate Global Optimization with Multiextremal Non-Differentiable Constraints Without Penalty Functions

This paper proposes a new algorithm for solving constrained global optimization problems where both the objective function and constraints are one-dimensional non-differentiable multiextremal Lipschitz functions. Multiextremal constraints can lead to complex feasible regions being collections of isolated points and intervals having positive lengths. The case is considered where the order the constraints are evaluated is fixed by the nature of the problem and a constraint i is defined only over the set where the constraint i−1 is satisfied. The objective function is defined only over the set where all the constraints are satisfied. In contrast to traditional approaches, the new algorithm does not use any additional parameter or variable. All the constraints are not evaluated during every iteration of the algorithm providing a significant acceleration of the search. The new algorithm either finds lower and upper bounds for the global optimum or establishes that the problem is infeasible. Convergence properties and numerical experiments showing a nice performance of the new method in comparison with the penalty approach are given. This research was supported by the following grants: FIRB RBNE01WBBB, FIRB RBAU01JYPN, and RFBR 04-01-00455-a. The author thanks Prof. D. Grimaldi for proposing the application discussed in the paper. An erratum to this article is available at .     

Optimization of upper semidifferentiable functions

In this paper, we present an implementable algorithm to minimize a nonconvex, nondifferentiable function in ^m . The method generalizes Wolfe's algorithm for convex functions and Mifflin's algorithm for semismooth functions to a broader class of functions, so-called upper semidifferentiable. With this objective, we define a new enlargement of Clarke's generalized gradient that recovers, in special cases, the enlargement proposed by Goldstein. We analyze the convergence of the method and discuss some numerical experiments.The author would like to thank J. B. Hiriart-Urruty (Toulouse) for having provided him with Definition 2.1 and the referees for their constructive remarks about a first version of the paper.     

Extension of the Frank-Wolfe algorithm to concave nondifferentiable objective functions

In this paper, we provide the theoretical foundations of an extension of the Frank-Wolfe algorithm for concave continuously differentiable objective functions to concave nondifferentiable objective functions, using a subgradient approach. The main theorem extends the results of Frank and Wolfe. A key main assumption is examined in the context of some special cases via three lemmas. A comparison is made with other subgradient approaches, and consideration is given to the two subproblem optimizations arising in the modified algorithm.This work was carried out under the auspices of the University of Manchester, Manchester, England. The author thanks the referees for their very helpful comments on the original version.     

A Multivariate Global Optimization Using Linear Bounding Functions

Recently linear bounding functions (LBFs) were proposed and used to find -global minima. This paper presents an LBF-based algorithm for multivariate global optimization problems. The algorithm consists of three phases. In the global phase, big subregions not containing a solution are quickly eliminated and those which possibly contain the solution are detected. An efficient scheme for the local phase is developed using our previous local minimization algorithm, which is globally convergent with superlinear/quadratic rate and does not require evaluation of gradients and Hessian matrices. To ensure that the found minimizers are indeed the global solutions or save computation effort, a third phase called the verification phase is often needed. Under adequate conditions the algorithm finds the -global solution(s) within finite steps. Numerical testing results illustrate how the algorithm works, and demonstrate its potential and feasibility.     

Nonconvex functions and variational inequalities

In this paper, we study some properties of a class of nonconvex functions, called semipreinvex functions, which includes the classes of preinvex functions and arc-connected convex functions. It is shown that the minimum of an arcwise directionally differentiable semi-invex functions on a semi-invex set can be characterized by a class of variational inequalities, known as variational-like inequalities. We use the auxiliary principle technique to prove the existence of a solution of a variational-like inequality and suggest a novel iterative algorithm.     

The use of random weights for the training of multilayer networks of neurons with Heaviside characteristics

Artificial neural networks have, in recent years, been very successfully applied in a wide range of areas. A major reason for this success has been the existence of a training algorithm called backpropagation. This algorithm relies upon the neural units in a network having input/output characteristics that are continuously differentiable. Such units are significantly less easy to implement in silicon than are neural units with Heaviside (step-function) characteristics. In this paper, we show how a training algorithm similar to backpropagation can be developed for 2-layer networks of Heaviside units by treating the network weights (i.e., interconnection strengths) as random variables. This is then used as a basis for the development of a training algorithm for networks with any number of layers by drawing upon the idea of internal representations. Some examples are given to illustrate the performance of these learning algorithms.     

A Tree Algorithm for Isotropic Finite Elements on the Sphere

The earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0.4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modeling of geoscientifically relevant data on the sphere plays an important role. In this paper, we deal with isotropic kernel functions showing a local support (briefly called isotropic finite elements) for reconstructing square-integrable functions on the sphere. An essential tool is the concept of multiresolution analysis by virtue of the spherical up function. Because the up function is built by an infinite convolution product, we do not know an explicit representation of it. However, the tree algorithm for the multiresolution analysis based on the up functions can be formulated by convolutions of isotropic kernels of low-order polynomial structure. For these kernels, we are able to find an explicit representation, so that the tree algorithm can be implemented efficiently.     

On Markovian decision programming with recursive reward functions

In this paper, the infinite horizon Markovian decision programming with recursive reward functions is discussed. We show that Bellman's optimal principle is applicable for our model. Then, a sufficient and necessary condition for a policy to be optimal is given. For the stationary case, an iteration algorithm for finding a stationary optimal policy is designed. The algorithm is a generalization of Howard's [7] and Iwamoto's [3] algorithms.This research was supported by the National Natural Science Foundation of China.     

Decompositions of positive self-dual boolean functions

A coterie, which is used to realize mutual exclusion in distributed systems, is a family C of subsets such that any pair of subsets in C has at least one element in common, and such that no subset in C contains any other subset in C. Associate with a family of subsets C a positive Boolean function f_c such that f_c(x) = 1 if the Boolean vector x is equal to or greater than the characteristic vector of some subset in C, and 0 otherwise. It is known that C is a coterie if and only if f_c is dual-minor, and is a non-dominated (ND) coterie if and only if f_c is self-dual. We study in this paper the decomposition of a positive self-dual function into smaller positive self-dual functions, as it explains how to represent and how to construct the corresponding ND coterie. A key step is how to decompose a positive dual-minor function f into a conjunction of positive self-dual functions f₁,f₂,…, f_k. In addition to the general condition for this decomposition, we clarify the condition for the decomposition into two functions f₁, and f₂, and introduce the concept of canonical decomposition. Then we present an algorithm that determines a minimal canonical decomposition, and a very simple algorithm that usually gives a decomposition close to minimal. The decomposition of a general self-dual function is also discussed.     

带非线性不等式约束优化问题的信赖域算法

借助于KKT条件和NCP函数,提出了求解带非线性不等式约束优化问题的信赖域算法.该算法在每一步迭代时,不必求解带信赖域界的二次规划子问题,仅需求一线性方程组系统.在适当的假设条件下,它还是整体收敛的和局部超线性收敛的.数值实验结果表明该方法是有效的.     

An RQP algorithm using a differentiable exact penalty function for inequality constrained problems

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.     

Factoring and recognition of read-once functions using cographs and normality and the readability of functions associated with partial k-trees

An approach for factoring general boolean functions was described in Golumbic and Mintz [Factoring logic functions using graph partitioning, in: Proceedings of IEEE/ACM International Conference on Computer Aided Design, November 1999, pp. 195-198] and Mintz and Golumbic [Factoring Boolean functions using graph partitioning, Discrete Appl. Math. 149 (2005) 131-153] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring read-once functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality.For non-read-once functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial k-tree, then F is a read 2^k function and a read 2^k formula for F can be obtained in polynomial time.     

等式约束非线性规划的一种线性代数方程组解法

本文提出了一种求解等式约束非线性规划的新方法－线性代数方程组求解方法。我们证明了该算法具有全局收敛性和局部二阶收敛性。     

Convergence of Cascade Algorithms in Sobolev Spaces for Perturbed Refinement Masks

In this paper, the convergence of the cascade algorithm in a Sobolev space is considered if the refinement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in terms of that of the masks.