Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive
Shaojun Zhang Zhong Wan
In this paper, a polymorphic uncertain nonlinear programming (PUNP) model is constructed to formulate the problem of maximizing the V-belt's fatigue life according to the practical engineering design conditions. The model is converted into an equivalent interval programming only involved with interval parameters for any given degree of membership and confidence level. Then, a deterministic equivalent formulation (DEF) for the original model is obtained based on the concept of possibility degree for the order of two interval numbers. An algorithm, called sampling based algorithm, is developed to find a robust optimal design scheme for maximizing the fatigue life of the V-belt. Case study is employed to demonstrate the validity and the practicability of the constructed model and the algorithm.
keywords: approximation-based optimal design. design optimization Belt drives polymorphic uncertainty modeling fatigue life
New approach to global minimization of normal multivariate polynomial based on tensor
Zhong Wan Chunhua Yang
In this paper, we first present a concise representation of multivariate polynomial, based on which we deduce the calculation formulae of its derivatives using tensor. Then, we propose a solution method to determine a global descent direction for the minimization of general normal polynomial. At a local and non-global maximizer or saddle point, we could use this method to get a global descent direction of the objective function. By using the global descent direction, we can transform an $n$-dimensional optimization problem into a one-dimensional one. Based on some efficient algorithms for one dimensional global optimization, we develop an algorithm to compute the global minimizer of normal multivariate polynomial. Numerical examples show that the proposed algorithm is promising.
keywords: polynomial optimization representation of polynomials global minimization. global descent direction tensor
A spectral PRP conjugate gradient methods for nonconvex optimization problem based on modified line search
Zhong Wan Chaoming Hu Zhanlu Yang
In this paper, a new spectral PRP conjugate gradient algorithm is developed for solving nonconvex unconstrained optimization problems. The search direction in this algorithm is proved to be a sufficient descent direction of the objective function independent of line search. To rule out possible unacceptably short step in the Armijo line search, a modified Armijo line search strategy is presented. The obtained step length is improved by employing the properties of the approximate Wolfe conditions. Under some suitable assumptions, the global convergence of the developed algorithm is established. Numerical experiments demonstrate that this algorithm is promising.
keywords: sufficiently descent direction line search Unconstrained optimization conjugate gradient global convergence.
Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem
Xiaoni Chi Zhongping Wan Zijun Hao
When there is uncertainty in the lower level optimization problem of a bilevel programming, it can be formulated by a robust optimization method as a bilevel program with lower level second-order cone programming problem (SOCBLP). In this paper, we show that the Lagrange multiplier set mapping of the lower level problem of a class of the SOCBLPs is upper semicontinuous under suitable assumptions. Based on this fact, we detect the similarities and relationships between the SOCBLP and its KKT reformulation. Then we derive the specific expression of the critical cone at a feasible point, and show that the second order sufficient conditions are sufficient for the second order growth at an M-stationary point of the SOCBLP under suitable conditions.
keywords: critical cone second order sufficient conditions. KKT reformulation upper semicontinuity Bilevel program with lower level second-order cone programming problem
A criterion for an approximation global optimal solution based on the filled functions
Liuyang Yuan Zhongping Wan Qiuhua Tang
In this paper, a new definition of the filled function is given. Based on the new definition, a new class of filled functions is constructed, and the properties of the new filled functions are analysed and discussed. Moreover, according to the new class of filled functions, a criterion is given to decide whether the point we have obtained is an approximate global optimal solution. Finally, a global optimization algorithm based on the new class of filled functions is presented. The implementation of the algorithm on several test problems is reported with numerical results.
keywords: a criterion local minimizer global minimizer. Global optimization the filled function
A filled function method for solving nonlinear complementarity problem
Liuyang Yuan Zhongping Wan Jingjing Zhang Bin Sun
In this paper a filled function method is suggested for solving the nonlinear complementarity problem. Firstly, the original problem is converted into a corresponding unconstrained optimization problem by using the Fischer-Burmeister function. Subsequently, a new filled function with one parameter is proposed for solving unconstrained optimization problems. Some properties of the filled function are studied and discussed without Lipschitz continuity condition. Finally, an algorithm based on the proposed filled function for solving the nonlinear complementarity problem is presented. The implementation of the algorithm on several test problems is reported with numerical results.
keywords: The nonlinear complementarity problem Local solution Global optimization Filled function method Global solution.
A new method for strong-weak linear bilevel programming problem
Yue Zheng Zhongping Wan Shihui Jia Guangmin Wang
We first propose an exact penalty method to solve strong-weak linear bilevel programming problem (for short, SWLBP) for every fixed cooperation degree from the follower. Then, we prove that the solution of penalized problem is also that of the original problem under some conditions. Furthermore, we give some properties of the optimal value function (as a function of the follower's cooperation degree) of SWLBP. Finally, we develop a method to acquire the critical points of the optimal value function without enumerating all values of the cooperation degree from the follower, and thus this function is also achieved. Numerical results show that the proposed methods are feasible.
keywords: partial cooperation pessimistic formulation penalty method critical point. optimistic formulation Bilevel programming
Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities
Jiawei Chen Zhongping Wan Liuyang Yuan
The notions of $\alpha$-well-posedness and generalized $\alpha$-well-posedness for a system of constrained variational inequalities involving set-valued mappings (for short, (SCVI)) are introduced in Hilbert spaces. Existence theorems of solutions for (SCVI) are established by using penalty techniques. Metric characterizations of $\alpha$-well-posedness and generalized $\alpha$-well-posedness, in terms of the approximate solutions sets, are presented. Finally, the equivalences between (generalized) $\alpha$-well-posedness for (SCVI) and existence and uniqueness of its solutions are also derived under quite mild assumptions.
keywords: existence of solution (generalized) $\alpha$-well-posedness metric characterization System of constrained set-valued variational inequalities $\alpha$-approximating sequence.
Sensitivity analysis in set-valued optimization under strictly minimal efficiency
Zhenhua Peng Zhongping Wan Weizhi Xiong

In this paper, the behavior of the perturbation map is analyzed quantitatively by virtue of contingent derivatives and generalized contingent epiderivatives for the set-valued maps under strictly minimal efficiency. The purpose of this paper is to provide some well-known results concerning sensitivity analysis by applying a separation theorem for convex sets. When the results regress to multiobjective optimization, some related conclusions are obtained in a multiobjective programming problem.

keywords: Sensitivity analysis perturbation maps contingent derivatives generalized contingent epiderivatives strictly minimal efficiency
Linear bilevel multiobjective optimization problem: Penalty approach
Yibing Lv Zhongping Wan

In this paper, we are interested by the linear bilevel multiobjective programming problem, where both the upper level and the lower level have multiple objectives. We approach this problem via an exact penalty method. Then, we propose an exact penalty function algorithm. Numerical results showing viability of the algorithm proposed are presented.

keywords: Linear bilevel multiobjective programming multiple objective programming pareto optimal solution penalty methods numerical results

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