Jianming Shi

Fractional Programming: the sum-of-ratios case ∗

One of the most difficult fractional programs encountered so far is the sum-of-ratios problem. Contrary to earlier expectations it is much more removed from convex programming than other multi-ratio problems analyzed before. It really should be viewed in the context of global optimization. It proves to be essentially [Formula: See Text]-hard in spite of its special structure under the usual assumptions on numerators and denominators. The article provides a recent survey of applications, theoretical results and various algorithmic approaches for this challenging problem.

Input–output substitutability and strongly monotonic p-norm least distance DEA measures

In DEA, there are two frameworks for efficiency assessment and targeting: the greatest and the least distance framework. The greatest distance framework provides us with the efficient targets that are determined by the farthest projections to the assessed decision making unit via maximization of the p-norm relative to either the strongly efficient frontier or the weakly efficient frontier. Non-radial measures belonging to the class of greatest distance measures are the slacks-based measure (SBM) and the range-adjusted measure (RAM). Whereas these greatest distance measures have traditionally been utilized because of their computational ease, least distance projections are quite often more appropriate than greatest distance projections from the perspective of managers of decision-making units because closer efficient targets may be attained with less effort. In spite of this desirable feature of the least distance framework, the least distance (in) efficiency versions of the additive measure, SBM and RAM do not even satisfy weak monotonicity. In this study, therefore, we introduce and investigate least distance p-norm inefficiency measures that satisfy strong monotonicity over the strongly efficient frontier. In order to develop these measures, we extend a free disposable set and introduce a tradeoff set that implements input–output substitutability.

Conical Partition Algorithm for Maximizing the Sum of dc Ratios

The following problem is considered in this paper:

Monotonicity of minimum distance inefficiency measures for Data Envelopment Analysis

max_{xin d{Sigma^m_{j=1}g_j(x)|h_j(x)},}, where,g_j(x)geq 0, and,h_j(x) > 0, j = 1,ldots,m,

New Generalized Invexity for Duality in Multiobjective Programming Problems Involving N-Set Functions

are d.c. (difference of convex) functions over a convex compact set D in R^n. Specifically, it is reformulated into the problem of maximizing a linear objective function over a feasible region defined by multiple reverse convex functions. Several favorable properties are developed and a branch-and-bound algorithm based on the conical partition and the outer approximation scheme is presented. Preliminary results of numerical experiments are reported on the efficiency of the proposed algorithm.

A trust-region algorithm for equality-constrained optimization via a reduced dimension approach

In this paper, we are concerned with a nondifferentiable minimax fractional programming problem. We derive a Kuhn-Tucker-type sufficient optimality condition for an optimal solution to the problem and establish week, strong and converse duality theorems for the problem and its three different forms of dual problems. The results in this paper extend a few known results in the literature.

MATHEMATICAL PROPERTIES OF DOMINANT AHP AND CONCURRENT CONVERGENCE METHOD

This research explores the minimum distance inefficiency measure for the Data Envelopment Analysis (DEA) model. A critical issue is that this measure does not satisfy monotonicity, i.e., the measure may provide a better evaluation score to an inferior decision making unit (DMU) than to a superior one. To overcome this, a variant called the extended facet approach has been introduced. This approach, however, requires a certain regularity condition to be met. We discuss several special classes of the DEA model, and show that for these models, the minimum distance inefficiency measure satisfies the monotonicity property without the regularity condition. Moreover, we conducted computational experiments using real-world data sets from these special classes, and demonstrated that the extended facet approach may overestimate the performance of a DMU.

Distance optimization approach to ratio-form efficiency measures in data envelopment analysis

In this paper, we discuss a kind of special nonconvex homogenous quadratic programming (HQP) and the methods to solve the HQP in an environment with certainty or uncertainty. In an environment with certainty, we first establish a strong duality between the HQP and its Lagrange dual problem, with the help of the fact that the Lagrange dual problem is equivalent to a convex semidefinite programming (SDP). Then we obtain a global solution to the HQP by solving the convex SDP. Furthermore, in an environment with uncertainty, we formulate the robust counterpart of the HQP to cope with uncertainty. We also establish the robust strong duality between the robust counterpart and its optimistic counterpart under a mild assumption. Since the counterpart is equivalent to a convex SDP under the same assumption, we can obtain a global solution to the robust counterpart by solving the convex SDP under the same assumption.