A GLOBALLY CONVERGENT QP-FREE ALGORITHM FOR INEQUALITY CONSTRAINED MINIMAX OPTIMIZATION

Although QP-free algorithms have good theoretical convergence and are effective in practice,their applications to minimax optimization have not yet been investigated.In this article,on the basis of the stationary conditions,without the exponential smooth function or constrained smooth transformation,we propose a QP-free algorithm for the nonlinear minimax optimization with inequality constraints.By means of a new and much tighter working set,we develop a new technique for constructing the sub-matrix in the lower right corner of the coefficient matrix.At each iteration,to obtain the search direction,two reduced systems of linear equations with the same coefficient are solved.Under mild conditions,the proposed algorithm is globally convergent.Finally,some preliminary numerical experiments are reported,and these show that the algorithm is promising.

SEMI-PROXIMAL POINT METHOD FOR NONSMOOTH CONVEX-CONCAVE MINIMAX OPTIMIZATION

Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas.While there have been many numerical algorithms for solving smooth convex-concave minimax problems,numerical algorithms for nonsmooth convex-concave minimax problems are rare.This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem.A semi-proximal point method(SPP)is proposed,in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem.This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen.We prove the global convergence of our algorithm under mild assumptions,without requiring strong convexity-concavity condition.Under the locally metrical subregularity of the solution mapping,we prove that our algorithm has the linear rate of convergence.Preliminary numerical results are reported to verify the efficiency of our algorithm.

Using Two-layer Minimax Optimization and DEA to Determine Attribute Weights

This study aims to explore a novel method for determining attribute weights,which is a key issue in constructing and analyzing multiple-attribute decision-making(MADM)problems.To this end,a hybrid approach combining the data envelopment analysis(DEA)model without explicit inputs(DEA-WEI)and a two-layer minimax optimization scheme is developed.It is demonstrated that in this approach,the most favorable set of weights is first considered for each alternative or decision-making unit(DMU)and these weight sets are then aggregated to determine the best compromise weights for attributes,with the interests of all DMUs simultaneously considered in a fair manner.This approach is particularly suitable for situations where the preferences of decision-makers(DMs)are either unclear or difficult to acquire.Two case studies are conducted to illustrate the proposed approach and its use for determining weights for attributes in practice.The first case concerns the assessment of research strengths of 24 selected countries using certain measures,and the second concerns the analysis of the performance of 64 selected Chinese universities,where the preferences of DMs are either unknown or ambiguous,but the weights of the attributes should be assigned in a fair and unbiased manner.

Optimal design of vibration absorber using minimax criterion with simplified constraints

In this paper, a minimax design of damped dynamic vibration absorber for a damped primary system is investigated to minimize the vibration magnitude peaks. Moreover, to reduce the sensitivity of the primary system response to variations of the forcing frequency for a two- degree-of-freedom system, the primary system should have two equal resonance magnitude peaks. To meet this re- quirement, a set of simplified constraint equations includ- ing distribution characteristics of the resonant frequencies of the primary system is established for the minimax objective function. The modified constraint equations have less un- known variables than those by other authors, which not only simplifies the computation but also improves the accuracy of the optimal values. The advantage of the proposed method is illustrated through numerical simulations.

Estimation of Parameters of Boundary Value Problems for Linear Ordinary Differential Equations with Uncertain Data

In this paper we construct optimal, in certain sense, estimates of values of linear functionals on solutions to two-point boundary value problems (BVPs) for systems of linear first-order ordinary differential equations from observations which are linear transformations of the same solutions perturbed by additive random noises. It is assumed here that right-hand sides of equations and boundary data as well as statistical characteristics of random noises in observations are not known and belong to certain given sets in corresponding functional spaces. This leads to the necessity of introducing minimax statement of an estimation problem when optimal estimates are defined as linear, with respect to observations, estimates for which the maximum of mean square error of estimation taken over the above-mentioned sets attains minimal value. Such estimates are called minimax mean square or guaranteed estimates. We establish that the minimax mean square estimates are expressed via solutions of some systems of differential equations of special type and determine estimation errors.

A NEW TRUST-REGION ALGORITHM FOR FINITE MINIMAX PROBLEM

In this paper, a new trust region algorithm for minimax optimization problems is proposed, which solves only one quadratic subproblem based on a new approximation model at each iteration. The approach is different with the traditional algorithms that usually require to solve two quadratic subproblems. Moreover, to avoid Maratos effect, the nonmonotone strategy is employed. The analysis shows that, under standard conditions, the algorithm has global and superlinear convergence. Preliminary numerical experiments are conducted to show the effiency of the new method.

Control efficiency optimization and Sobol＇s sensitivity indices of MTMDs design parameters for buffeting and flutter vibrations in a cable stayed bridge

This paper studies optimization of three design parameters （mass ratio, frequency ratio and damping ratio） of multiple tuned mass dampers MTMDs that are applied in a cable stayed bridge excited by a strong wind using minimax optimization technique. ABAQUS finite element program is utilized to run numerical simulations with the support of MATLAB codes and Fast Fourier Transform FFT technique. The optimum values of these three parameters are validated with two benchmarks from the literature, first with Wang and coauthors and then with Lin and coauthors. The validation procedure detected a good agreement between the results. Box-Behnken experimental method is dedicated to formulate the surrogate models to represent the control efficiency of the vertical and torsional vibrations. Sobol＇s sensitivity indices are calculated for the design parameters in addition to their interaction orders. The optimization results revealed better performance of the MTMDs in controlling the vertical and the torsional vibrations for higher mode shapes. Furthermore, the calculated rational effects of each design parameter facilitate to increase the control efficiency of the MTMDs in conjunction with the support of the surrogate models.