A continuous approch for globally solving linearly constrained quadratic

In a continuous approach we propose an efficient method for globally solving linearly constrained quadratic zero-one programming considered as a d.c. (difference of onvex functions) program. A combination of the d.c. optimization algorithm (DCA) which has a finite convergence, and the branch-and-bound scheme was studied. We use rectangular bisection in the branching procedure while the bounding one proceeded by applying d.c.algorithms from a current best feasible point (for the upper bound) and by minimizing a well tightened convex underestimation of the objective function on the current subdivided domain (for the lower bound). DCA generates a sequence of points in the vertex set of a new polytope containing the feasible domain of the problem being considered. Moreover if an iterate is integral then all following iterates are integral too.Our combined algorithm converges so quite often to an integer approximate solution.Finally, we present computational results of several test problems with up to 1800

variables which prove the efficiency of our method, in particular, for linear zero-one programming     

求解非凸区域上凸函数比式和问题的全局优化方法

针对非凸区域上的凸函数比式和问题,给出一种求其全局最优解的确定性方法.该方法基于分支定界框架.首先通过引入变量,将原问题等价转化为d.c.规划问题,然后利用次梯度和凸包络构造松弛线性规划问题,从而将关键的估计下界问题转化为一系列线性规划问题,这些线性规划易于求解而且规模不变,更容易编程实现和应用到实际中;分支采用单纯形对分不但保证其穷举性,而且使得线性规划规模更小.理论分析和数值实验表明所提出的算法可行有效.     

Improving the efficiency of DC global optimization methods by improving the DC representation of the objective function

There are infinitely many ways of representing a d.c. function as a difference of convex functions. In this paper we analyze how the computational efficiency of a d.c.optimization algorithm depends on the representation we choose for the objective function, and we address the problem of characterizing and obtaining a computationally optimal representation. We introduce some theoretical concepts which are necessary for this analysis and report some numerical experiments.      

A branch and reduce approach for solving a class of low rank d.c. programs

Various classes of d.c. programs have been studied in the recent literature due to their importance in applicative problems. In this paper we consider a branch and reduce approach for solving a class of d.c. problems. Seven partitioning rules are analyzed and some techniques aimed at improving the overall performance of the algorithm are proposed. The results of a computational experience are provided in order to point out the performance effectiveness of the proposed techniques.     

Optimization of a Long-Short Portfolio under Nonconvex Transaction Cost

The purpose of this paper is to propose a practical branch and bound algorithm for solving a class of long-short portfolio optimization problem with concave and d.c. transaction cost and complementarity conditions on the variables.We will show that this algorithm can solve a problem of practical size and that the long-short strategy leads to a portfolio with significantly better risk-return structure compared with standard purchase only portfolio both in terms of ex-ante and ex-post performance.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

A D.C. biobjective location model

In this paper we address the biobjective problem of locating a semiobnoxious facility, that must provide service to a given set of demand points and, at the same time, has some negative effect on given regions in the plane. In the model considered, the location of the new facility is selected in such a way that it gives answer to these contradicting aims: minimize the service cost (given by a quite general function of the distances to the demand points) and maximize the distance to the nearest affected region, in order to reduce the negative impact. Instead of addressing the problem following the traditional trend in the literature (i.e., by aggregation of the two objectives into a single one), we will focus our attention in the construction of a finite -dominating set, that is, a finite feasible subset that approximates the Pareto-optimal outcome for the biobjective problem. This approach involves the resolution of univariate d.c. optimization problems, for each of which we show that a d.c. decomposition of its objective can be obtained, allowing us to use standard d.c. optimization techniques.     

Solving a Class of Linearly Constrained Indefinite Quadratic Problems by D.C. Algorithms

Linearly constrained indefinite quadratic problems play an important role in global optimization. In this paper we study d.c. theory and its local approachto such problems. The new algorithm, CDA, efficiently produces local optima and sometimes produces global optima. We also propose a decomposition branch andbound method for globally solving these problems. Finally many numericalsimulations are reported.     

Portfolio optimization under D.C. transaction costs and minimal transaction unit constraints

This paper addresses itself to a portfolio optimization problem under nonconvex transaction costs and minimal transaction unit constraints. Associated with portfolio construction is a fee for purchasing assets. Unit transaction fee is larger when the amount of transaction is smaller. Hence the transaction cost is usually a concave function up to certain point. When the amount of transaction increases, the unit price of assets increases due to illiquidity/market impact effects. Hence the transaction cost becomes convex beyond certain bound. Therefore, the net expected return becomes a general d.c. function (difference of two convex functions). We will propose a branch-and-bound algorithm for the resulting d.c. maximization problem subject to a constraint on the level of risk measured in terms of the absolute deviation of the rate of return of a portfolio. Also, we will show that the minimal transaction unit constraints can be incorporated without excessively increasing the amount of computation.     

An Inverse Coefficient Problem for an Integro-differential Equation

In this article we consider the inverse coefficient problem of recovering the function { ( x ) system of partial differential equations that can be reduced to a second order integro-differential equation $ -u_{xx} + c(x)u_{x} + d\phi (x)u-\gamma d\phi (x)\int _{0}^{t}e^{-\gamma (t-\tau )}u(x,\tau )\, d\tau = 0 $ with boundary conditions. We prove the existence and uniqueness of solutions to the inverse problem and develop a numerical algorithm for solving this problem. Computational results for some examples are presented.     

Generalized Quasilinearization for Quasilinear Parabolic Equations with Nonlinearities of DC Type

In this paper, we consider an initial boundary-value problem for a class of quasilinear parabolic equations whose lower-order nonlinearity is of d.c. function type with respecxt to the dependent variable. Assuming the existence of an ordered pair of weak upper and lower solutions, we establish a generalized quasilinearization method for the problem under consideration. A characteristic feature of this generalized quasilinearization method consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic.     

Branch-and-bound approaches to standard quadratic optimization problems

This paper explores several possibilities for applying branch-and-bound techniques to a central problem class in quadratic programming, the so-called Standard Quadratic Problems (StQPs), which consist of finding a (global) minimizer of a quadratic form over the standard simplex. Since a crucial part of the procedures is based on efficient local optimization, different procedures to obtain local solutions are discussed, and a new class of ascent directions is proposed, for which a convergence result is established. Main emphasis is laid upon a d.c.-based branch-and-bound algorithm, and various strategies for obtaining an efficient d.c. decomposition are discussed.     

D.C. programming approach for multicommodity network optimization problems with step increasing cost functions

We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows x ^* such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(x^*)-LB)/{f(x^*)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.     

On the minimum corridor connection problem and other generalized geometric problems

In this paper we discuss the complexity and approximability of the minimum corridor connection problem where, given a rectilinear decomposition of a rectilinear polygon into “rooms”, one has to find the minimum length tree along the edges of the decomposition such that every room is incident to a vertex of the tree. We show that the problem is strongly NP-hard and give a subexponential time exact algorithm. For the special case when the room connectivity graph is k-outerplanar the algorithm running time becomes cubic. We develop a polynomial time approximation scheme for the case when all rooms are fat and have nearly the same size. When rooms are fat but are of varying size we give a polynomial time constant factor approximation algorithm.     

Improving the algorithm of Bafna and Pevzner for the problem of sorting by transpositions: a practical approach

In computational biology, genome rearrangements is a field in which we investigate the combinatorial problem of sorting by transpositions. This problem consists in finding the minimum number of transpositions (mutational event) that transform a chromosome into another. Bafna and Pevzner [SIAM J. 11 (2) (1998) 224–240] proposed a 1.5-approximation algorithm to solve this problem, using a structure called cycle graph. In this work, we first present results that allowed us to implement their algorithm, maintaining the 1.5-approximation ratio. The present implementation runs in O(n³) time complexity, noting that we created a data structure to store the cycle graph in memory in O(n) time complexity. The results obtained from the program allowed us to propose heuristics, that further improved the performance of the original algorithm. Comparing our experimental results with the best results published so far, we achieved better performance. Besides, we developed a program to visualize the cycle graphs and the transpositions indicated by the algorithm. This work targets to contribute for discovering the complexity of the problem of sorting by transpositions, which remains open.     

A Branch and Bound Method via d.c. Optimization Algorithms and Ellipsoidal Technique for Box Constrained Nonconvex Quadratic Problems

In this paper we propose a new branch and bound algorithm using a rectangular partition and ellipsoidal technique for minimizing a nonconvex quadratic function with box constraints. The bounding procedures are investigated by d.c. (difference of convex functions) optimization algorithms, called DCA. This is based upon the fact that the application of the DCA to the problems of minimizing a quadratic form over an ellipsoid and/or over a box is efficient. Some details of computational aspects of the algorithm are reported. Finally, numerical experiments on a lot of test problems showing the efficiency of our algorithm are presented.     

A Finite Algorithm for a Particular D.C. Quadratic Programming Problem

In this paper a particular quadratic minimum program, having a particular d.c. objective function, is studied. Some theoretical properties of the problem are stated and the existence of minimizers is characterized. A solution algorithm, based on the so called optimal level solutions approach, is finally proposed.     

A new iterative method for the split common fixed point problem in Hilbert spaces

The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed-point set. In this paper, we propose a new algorithm for this problem that is completely different from the existing algorithms. Moreover, our algorithm does not need any prior information of the operator norm. Under standard assumptions, we establish a weak convergence theorem of the proposed algorithm and a strong convergence theorem of its variant.     

D.c. sets,d.c. functions and nonlinear equations

Ad.c. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in ⁿ into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations—inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.The revision of this paper was produced during the author's stay supported by a Sophia lecturing-research grant at Sophia University (Tokyo, Japan).