New approach to global minimization of normal multivariate polynomial based on tensor
Zhong Wan Chunhua Yang
Journal of Industrial & Management Optimization 2008, 4(2): 271-285 doi: 10.3934/jimo.2008.4.271
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
Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive
Shaojun Zhang Zhong Wan
Journal of Industrial & Management Optimization 2012, 8(2): 493-505 doi: 10.3934/jimo.2012.8.493
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
A spectral PRP conjugate gradient methods for nonconvex optimization problem based on modified line search
Zhong Wan Chaoming Hu Zhanlu Yang
Discrete & Continuous Dynamical Systems - B 2011, 16(4): 1157-1169 doi: 10.3934/dcdsb.2011.16.1157
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.
An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints
Jiao-Yan Li Xiao Hu Zhong Wan
Journal of Industrial & Management Optimization 2017, 13(5): 1-18 doi: 10.3934/jimo.2018200

Vehicle routing problem (VRP) is a typical and important combinatorial optimization problem, and is often involved with complicated temporal and spatial constraints in practice. In this paper, the VRP is formulated as an optimization model for minimizing the number of vehicles and the total transportation cost subject to constraints on loading plan, service time and weight capacity. The transportation cost consists of the rent charge of vehicles, fuel cost, and carbon tax. Owing to complexity of the built model, it is divided into two subproblems by a two-stage optimization approach: at the first stage, the number of vehicles is minimized, then the routing plan is optimized at the second stage. For solving the sequential subproblems, two correlated genetic algorithms are developed, which share the same initial population to reduce their computational costs. Numerical results indicate that the developed algorithms are efficient, and a number of important managerial insights are revealed from the model.

keywords: Vehicle routing problems optimization model algorithms carbon emissions bi-objective optimization

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