Wenxun Xing

Parallel machine scheduling with splitting jobs

Abstract To schedule n jobs on m parallel machines with the minimum total cost is the parallel machine scheduling (PMS) problem. Generally, there is a hypothesis: a job cannot be processed on two machines simultaneously if preemption is allowed. When the processing requirement of a job is considered as the demand of a product, jobs can be split arbitrarily to continuous sublots and processed independently on m machines. So, we can discuss PMS under a hypothesis: any part of a job can be processed on two different machines at the same time, and we call it PMS with splitting jobs. In this paper, we first present some simple cases which are polynomial solvable. Furthermore, a heuristic ML and its worst-case analysis are shown for P / split / C max with independent job setup times. The worst-case performance ratio of ML is within 7 4 −1/m (m⩾2) .

Global optimization for a class of fractional programming problems

This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the “problem-defining” matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.

KKT Solution and Conic Relaxation for Solving Quadratically Constrained Quadratic Programming Problems

To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature. Moreover, we propose a computational scheme that provides clues of designing effective algorithms for more solvable quadratically constrained quadratic programming problems.

New trends in parallel machine scheduling

Reviews some new trends in parallel machine scheduling (PMS). PMS, as an area of research, is governed by questions that arise in production planning, flexible manufacture systems, computer control, etc. The main characteristic of these problems is to optimize an objective, with jobs to be finished on a series of machines with the same function. Gives a short review of new developments in PMS associated with the problems of just‐in‐time (JIT) production, pre‐emption with set‐up, and capacitated machine scheduling. Discusses non‐regular objectives oriented by the JIT concept; pre‐emption with set‐up; capacitated machine scheduling; and relationships between PMS and vehicle routeing problems.

No-wait flexible flowshop scheduling with no-idle machines

This paper considers a two-stage flexible flowshop scheduling problem with no waiting time between two sequential operations of a job and no idle time between two consecutive processed jobs on machines of the second stage. We show its complexity and present a heuristic algorithm with asymptotically tight error bounds.

Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme

It is co-NP-complete to decide whether a given matrix is copositive or not. In this paper, this decision problem is transformed into a quadratic programming problem, which can be approximated by solving a sequence of linear conic programming problems defined on the dual cone of the cone of nonnegative quadratic functions over the union of a collection of ellipsoids. Using linear matrix inequalities (LMI) representations, each corresponding problem in the sequence can be solved via semidefinite programming. In order to speed up the convergence of the approximation sequence and to relieve the computational effort of solving linear conic programming problems, an adaptive approximation scheme is adopted to refine the union of ellipsoids. The lower and upper bounds of the transformed quadratic programming problem are used to determine the copositivity of the given matrix.

On the global optimality of generalized trust region subproblems

Quadratically constrained quadratic programming is an important class of optimization problems. We consider the case with one quadratic constraint. Since both the objective function and its constraint can be neither convex nor concave, it is also known as the ‘generalized trust region subproblem.’ The theory and algorithms for this problem have been well studied under the Slater condition. In this article, we analyse the duality property between the primal problem and its Lagrangian dual problem, and discuss the attainability of the optimal primal solution without the Slater condition. The relations between the Lagrangian dual and semidefinite programming dual is also given.

A bin packing problem with over-sized items

This paper considers a variable-sized bin packing problem with over-sized items, where some item sizes are larger than the largest size of bins. In practice, two-stage procedures are used to handle this problem. The first stage is to pack each over-sized item into the largest bins fully, and the second stage is to pack the remaining parts of the over-sized items and non-over-sized items using the methods of variable-sized bin packing. We analyze two-stage procedures in a worst case version and find that the procedures have no better worst case ratios than 2. Finally we give an on-line algorithm with an asymptotic worst case ratio no worse than 74.

Adaptive computable approximation to cones of nonnegative quadratic functions

Cones of nonnegative quadratic functions are keys to the understanding of quadratic optimization problems, since any quadratically constrained quadratic programming problem can be reformulated as a linear conic programming problem over such a cone. This paper proposes an adaptive computable approximation scheme to cones of nonnegative quadratic functions and uses it for solving linear conic programming problems over such a cone. We study some basic properties of cones of nonnegative quadratic functions and present a class of simple cones with computable linear matrix inequalities representations. Building on these simple cones, we design a computable approximation scheme for handling a general cone of nonnegative quadratic functions. When the scheme is applied for solving linear conic programming problems over a cone of nonnegative quadratic functions, we incorporate an adaptive approach to enhance the performance of the proposed computable approximation scheme. The computational performance and theoretic convergence proof of the proposed adaptive computable approximation scheme are shown for solving box-constrained quadratic programming problems.

An improved lower bound and approximation algorithm for binary constrained quadratic programming problem

This paper presents an improved lower bound and an approximation algorithm based on spectral decomposition for the binary constrained quadratic programming problem. To decompose spectrally the quadratic matrix in the objective function, we construct a low rank problem that provides a lower bound. Then an approximation algorithm for the binary quadratic programming problem together with a worst case performance analysis for the algorithm is provided.