A note on the method of multipliers

Implementation of the penalty function method for constrained optimization poses numerical difficulties as the penalty parameter increases. To offset this problem, one often resorts to Newton's method. In this note, working in the context of the penalty function method, we establish an intimate connection between the second-order updating formulas which result from Newton's method on the primal problem and Newton's method on the dual problem.The author wishes to thank Professor R. A. Tapia for his careful review of this note. He has contributed significantly to its content through several crucial observations.     

Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems

We consider the following optimization problem: in an abstract setX, find and elementx that minimizes a real functionf subject to the constraintsg(x)0 andh(x)=0, whereg andh are functions fromX into normed vector spaces. Assumptions concerning an overall convex structure for the problem in the image space, the existence of interior points in certain sets, and the normality of the constraints are formulated. A theorem of the alternative is proved for systems of equalities and inequalities, and an intrinsic multiplier rule and a Lagrangian saddle-point theorem (strong duality theorem) are obtained as consequences.     

A method for member selection of cross-functional teams using the individual and collaborative performances

The member selection problem is an important aspect of the formation of cross-functional teams (CFTs). Selecting suitable members from a set of candidates will facilitate the successful task accomplishment. In the existing studies of member selection, the individual performance concerning a single candidate is mostly used, whereas the collaborative performance associating with a pair of candidates is overlooked. In this paper, as a solution to this problem, we propose a method for member selection of CFTs, where both the individual performance of candidates and the collaborative performance between candidates are considered. In order to select the desired members, firstly, a multi-objective 0–1 programming model is built using the individual and collaborative performances, which is an NP-hard problem. To solve the model, we develop an improved nondominated sorting genetic algorithm II (INSGA-II). Furthermore, a real example is employed to illustrate the suitability of the proposed method. Additionally, extensive computational experiments to compare INSGA-II with the nondominated sorting genetic algorithm II (NSGA-II) are conducted and much better performance of INSGA-II is observed.