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OPTIMALITY CONDITIONS IN NONSMOOTH MULTIOBJECTIVE PROGRAMMING
非光滑多目标规划的最优性条件

LIU SAN-YANG,
刘三阳

系统科学与数学 , 1989,
Abstract: In this paper,necessary and sufficient optimality conditions in nonsmooth multiobjectiveprogramming are established with the help of the upper right Dini-derivative.
Several Guaranteed Descent Conjugate Gradient Methods for Unconstrained Optimization
San-Yang Liu,Yuan-Yuan Huang
Journal of Applied Mathematics , 2014, DOI: 10.1155/2014/825958
Abstract: This paper investigates a general form of guaranteed descent conjugate gradient methods which satisfies the descent condition and which is strongly convergent whenever the weak Wolfe line search is fulfilled. Moreover, we present several specific guaranteed descent conjugate gradient methods and give their numerical results for large-scale unconstrained optimization. 1. Introduction Consider the following unconstrained optimization problem: where is the -dimensional Euclidean space, is continuously differentiable, and its gradient is available. Conjugate gradient methods are very efficient to solve problem (1) due to their simple iteration and their low memory requirements. For any given starting point , they generate a sequence by the following recursive relation: where , is a step length obtained by means of a one-dimensional search, and is a scalar that characterizes the method. In general, the step length in (2) is obtained by fulfilling the following weak Wolfe conditions [1, 2]: where . And different choices for the scalar in (3) result in different nonlinear conjugate gradient methods. Well-known formulas for are the Fletcher-Reeves (FR), Hestenes-Stiefel (HS), Polak-Ribiére-Polyak (PRP), Dai-Yuan (DY), and Liu-Storey (LS) formulas (see [3], [4], [5], [6], [7], and [8], resp.) and are given by where means the Euclidean norm and . Their corresponding conjugate gradient methods are viewed as basic conjugate gradient methods. Among these basic conjugate gradient methods, the PRP and HS methods perform very similarly and perform better than other basic conjugate gradient methods [9]. While Powell [10] utilized an example illustrating that the PRP and HS methods may cycle without approaching any solution point, then modified versions of the PRP and HS methods were presented by many researchers (see, e.g., [11–16]). The following (sufficient) descent condition, is very important for conjugate gradient methods, so we are particularly interested in the conjugate gradient methods with sufficient descent conditions. Up to now, there are many descent conjugate gradient methods proposed by researchers; please see [12, 16–19] and references therein. One well-known guaranteed descent conjugate gradient method was proposed by Hager and Zhang [16, 20, 21] with The method is designed based on the HS method and satisfies the sufficient descent condition (6) with for any (inexact) line search. In [18], Zhang and Li proposed a general case of the HZ method with where and is a scalar to be specified. It also satisfies the sufficient descent condition (6) with , and
Existence of Traveling Fronts in a Food-Limited Population Model with Spatiotemporal Delay
Hai-Qin Zhao,San-Yang Liu
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/705197
Abstract: This paper is concerned with the traveling fronts of a diffusive food-limited population model with spatiotemporal delay. Sufficient conditions are established for the existence of traveling wave fronts by choosing different kinds of delay kernels. The approach used here is the upper-lower solution method and monotone iteration technique. Our work extends and/or covers some previous results. 1. Introduction This paper is concerned with the traveling fronts for the following food-limited model: where , , and are nonnegative constants, , and the kernel is any integrable nonnegative function satisfying , which was first proposed and analyzed by Gourley and So [1] on a finite domain . In the case , , , (1.1) becomes Recently, many researchers studied the existence of traveling fronts of (1.3) with some specific . For the case where is the Dirac delta function, Gourley [2] showed that, for any , there exists such that, for any , (1.3) has a traveling front connecting the equilibria and , by using the approach developed by Wu and Zou [3]. For the case Gourley and Chaplain [4] proved the existence of traveling fronts for any and sufficient small , by employing linear chain techniques to recast the traveling wave equations as a finite-dimensional system of ODEs and using Fenichel's geometric singular perturbation theory [5] and the Fredholm alternative. For the case Gourley and Chaplain [4], by using the method of Canosa [6], obtained some information on the monotonicity of traveling fronts for sufficiently large . Furthermore, for these cases Wang and Li [7] showed that, for any , there exists (or ) such that for any (or ), (1.3) has a traveling front connecting the equilibria and . In this paper, based on the monotone iteration technique as well as the upper and lower solution method developed by Wang et al. [8], we will establish the existence of traveling fronts of (1.1) with the kernel functions (1.4)–(1.7). More precisely, we shall show that for any , there exists (or ) such that, for any (or ), (1.1) has a traveling front connecting the equilibria and (see Theorems 2.5 and 2.9 and Remark 2.10), which includes, improves, and/or complements a number of existing results in [2–4, 7, 9, 10]. The rest of the paper is organized as follows. In Section 2, we establish the existence of traveling wave fronts of (1.1) with the kernel functions (1.4)–(1.7). For the sake of convenience, we present in the Appendix some results developed by Wang et al. [8]. 2. Existence of Traveling Fronts In this section, we will use Theorem A.2 to establish the existence of traveling
A New Global Optimization Algorithm for Solving Generalized Geometric Programming
San-Yang Liu,Chun-Feng Wang,Li-Xia Liu
Mathematical Problems in Engineering , 2010, DOI: 10.1155/2010/346965
Abstract: A global optimization algorithm for solving generalized geometric programming (GGP) problem is developed based on a new linearization technique. Furthermore, in order to improve the convergence speed of this algorithm, a new pruning technique is proposed, which can be used to cut away a large part of the current investigated region in which the global optimal solution does not exist. Convergence of this algorithm is proved, and some experiments are reported to show the feasibility of the proposed algorithm. 1. Introduction This paper considers generalized geometric programming GGP problem in the following form: where . Generally speaking, GGP problem is a non convex programming problem, which has a wide variety of applications, such as in engineering design, economics and statistics, manufacturing, and distribution contexts in risk management problems [1–4]. During the past years, many local optimization approaches for solving GGP problem have been presented [5, 6], but global optimization algorithms based on the characteristics of GGP problem are scarce. Maranas and Floudas [7] proposed a global optimization algorithm for solving GGP problem based on convex relaxation. Shen and Zhang [8] presented a method to globally solve GGP problem by using linear relaxation. Recently, several branch and bound algorithms have been developed [9, 10]. The purpose of this paper is to introduce a new global optimization algorithm for solving GGP problem. In this algorithm, by utilizing the special structure of GGP problem, a new linear relaxation technique is presented. Based on this technique, the initial GGP problem is systematically converted into a series of linear programming problems. The solutions of these converted problems can approximate the global optimal solution of GGP problem by successive refinement process. The main features of this algorithm are: (1) a new linearization technique for solving GGP problem is proposed, which applies more information of the functions of GGP problem, (2) the generated relaxation linear programming problems are embedded within a branch and bound algorithm without increasing the number of variables and constraints, (3) a new pruning technique is presented, which can be used to improve the convergence speed of the proposed algorithm, and (4) numerical experiments are given, which show that the proposed algorithm can treat all of the test problems in finding global optimal solution within a prespecified tolerance. The structure of this paper is as follows. In Section 2, first, we construct the lower approximate linear functions
Multi-construction Ant Colony Optimization Algorithm for Permutation Flow Shop Scheduling
多构造蚁群优化求解置换流水车间调度问题

LIU Yan-feng,LIU San-yang,
刘延风
,刘三阳

计算机科学 , 2010,
Abstract: A Multi Construction Ant Colony Optimization Algorithm for Permutation Flow Shop Scheduling was proposed.In this algorithm,solutions are constructed through two modes,which are based on Nawaz-Enscore-Ham heuristics and Rajendran heuristics respectively.Then the proportion of construction modes is adjusted adaptively according to quality of solution constructed.Simulation results and comparisons based on benchmarks demonstrate the effectiveness of the algorithm.
Algorithm Based on Genetic Algorithm for Sudoku Puzzles
基于遗传算法求解数独难题

LIU Yan-feng,LIU San-yang,
刘延风
,刘三阳

计算机科学 , 2010,
Abstract: To solve the Sudoku puzzles,above all,they were changed into a combinatorial optimization problem.Then,a genetic algorithm with specialized encoding,initialization and local search operator was presented to optimize it.The experimental results show the algorithm is effective for all difficulty levels Sudoku puzzles.
Permutation flow shop scheduling algorithm based on ant colony optimization
置换流水车间调度的蚁群优化算法

LIU Yan-feng,LIU San-yang,
刘延风
,刘三阳

计算机应用 , 2008,
Abstract: An ant colony optimization based algorithm for permutation flow shop scheduling was proposed. The key point of this algorithm is to integrate NEH heuristics with ant colony optimization. The proposed algorithm was tested on scheduling problem benchmarks. Theoretical analysis and experimental results show its effectiveness.
A Branch-and-Reduce Approach for Solving Generalized Linear Multiplicative Programming
Chun-Feng Wang,San-Yang Liu,Geng-Zhong Zheng
Mathematical Problems in Engineering , 2011, DOI: 10.1155/2011/409491
Abstract: We consider a branch-and-reduce approach for solving generalized linear multiplicative programming. First, a new lower approximate linearization method is proposed; then, by using this linearization method, the initial nonconvex problem is reduced to a sequence of linear programming problems. Some techniques at improving the overall performance of this algorithm are presented. The proposed algorithm is proved to be convergent, and some experiments are provided to show the feasibility and efficiency of this algorithm.
Existence of Feasible Solutions of Bilevel Programming
二层规划可行解的存在性

FAN Li-Ya,LIU San-Yang,
范丽亚
,刘三阳

数学物理学报(A辑) , 2003,
Abstract: Bilevel programming is usually formulated with two optimization problems where the constraint set of the first one (upper level problem) is partially determined by the optimal reactions of the second one (lower level problem). The paper is devoted to study the existence of feasible solutions, which plays a fundamental and important role in bilevel programming, by means of the w pseudo monotonicity of the Clarke's subdifferential mapping of the lower level objective function.
Complete Space like Submanifolds with Flat Connection of Normal Bundle in the de Sitter Space
de Sitter空间中法联络平坦的完备类空子流形

SHU Shi-Chang,LIU San-Yang,
舒世昌
,刘三阳

数学物理学报(A辑) , 2004,
Abstract: Let M be a complete n -demensional space-like submanifold in the de Sitter space Sp(n+p)(c) with parallel mean curvature vector and constant scalar curvature, if theconnection of normal bundle is flat and K(M)<0, or then M is atotally umbilical submanifold.
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