## SEARCH

#### Author

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- Wu, Weili 6 (%)
- Butenko, Sergiy 5 (%)
- Khanh, Phan Quoc 5 (%)
- Pardalos, Panos M. 5 (%)
- Chuang, Chih-Sheng 4 (%)

#### Subject

- Computer Science, general 384 (%)
- Economics / Management Science [x] 384 (%)
- Operations Research/Decision Theory 384 (%)
- Optimization 384 (%)
- Real Functions 384 (%)

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## A black-box scatter search for optimization problems with integer variables

### Journal of Global Optimization (2014-03-01) 58: 497-516 , March 01, 2014

The goal of this work is the development of a black-box solver based on the scatter search methodology. In particular, we seek a solver capable of obtaining high quality outcomes to optimization problems for which solutions are represented as a vector of integer values. We refer to these problems as integer optimization problems. We assume that the decision variables are bounded and that there may be constraints that require that the black-box evaluator is called in order to know whether they are satisfied. Problems of this type are common in operational research areas of applications such as telecommunications, project management, engineering design and the like.Our experimental testing includes 171 instances within four classes of problems taken from the literature. The experiments compare the performance of the proposed method with both the best context-specific procedures designed for each class of problem as well as context-independent commercial software. The experiments show that the proposed solution method competes well against commercial software and that can be competitive with specialized procedures in some problem classes.

## Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

### Journal of Global Optimization (2014-08-01) 59: 695-727 , August 01, 2014

This paper aims at showing that the class of augmented Lagrangian functions for nonlinear semidefinite programming problems can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem. By means of the image space analysis, a global saddle point condition for the augmented Lagrangian function is investigated. It is shown that the existence of a saddle point is equivalent to a regular nonlinear separation of two suitable subsets of the image space. Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set. Motivated by the result on global saddle points, we propose two modified primal-dual methods based on the augmented Lagrangian using different strategies and prove their convergence to a global solution and the optimal value of the original problem without requiring the boundedness condition of the multiplier sequence.

## A full-Newton step feasible interior-point algorithm for $$P_*(\kappa )$$ P ∗ ( κ ) -linear complementarity problems

### Journal of Global Optimization (2014-05-01) 59: 81-99 , May 01, 2014

In this paper, a full-Newton step feasible interior-point algorithm is proposed for solving $$P_*(\kappa )$$ -linear complementarity problems. We prove that the full-Newton step to the central path is local quadratically convergent and the proposed algorithm has polynomial iteration complexity, namely, $$O\left( (1+4\kappa )\sqrt{n}\log {\frac{n}{\varepsilon }}\right) $$ , which matches the currently best known iteration bound for $$P_*(\kappa )$$ -linear complementarity problems. Some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithm.

## On cyclic and $$n$$ n -cyclic monotonicity of bifunctions

### Journal of Global Optimization (2014-12-01) 60: 599-616 , December 01, 2014

In the recent literature, the connection between maximal monotone operators and the Fitzpatrick function is investigated. Subsequently, this relation has been extended to maximal monotone bifunctions and their Fitzpatrick transform. In this paper we generalize some of these results to maximal $$n$$ -cyclically monotone and maximal cyclically monotone bifunctions, by introducing and studying the Fitzpatrick transforms of order $$n$$ or infinite order for bifunctions.

## Best proximity point theorems: exposition of a significant non-linear programming problem

### Journal of Global Optimization (2013-08-01) 56: 1699-1705 , August 01, 2013

The primary goal of this work is to address the non-linear programming problem of globally minimizing the real valued function *x* → *d*(*x*, *Tx*) where *T* is presumed to be a non-self mapping that is a generalized proximal contraction in the setting of a metric space. Indeed, an iterative algorithm is presented to determine a solution of the preceding non-linear programming problem that focuses on global optimization. As a sequel, one can compute optimal approximate solutions to some fixed point equations and optimal solutions to some unconstrained non-linear programming problems.

## System modeling and performance analysis of the power saving class type II in BWA networks

### Journal of Global Optimization (2013-08-01) 56: 1375-1391 , August 01, 2013

For reducing the energy consumption of the Mobile Station in mobile Broadband Wireless Access networks, IEEE 802.16 offers three kinds of sleep mode operations called power saving classes type I, type II and type III. In order to investigate mathematically the inherent relationships between the performance measures and the system parameters, we propose in this paper a novel method for modeling the sleep mode with the power saving class type II in IEEE 802.16 and analyzing the performance of this sleep mode. Considering the attractive feature that some data frames can be transmitted during the listening state, we present a queueing model with two kinds of busy mechanisms to capture the working principle of the sleep mode operations with the power saving class type II. With the first and higher derivatives of the probability generating functions, we can give the averages and the standard deviations for the system performance using the diffusion approximation for the operating process of the system. We also propose methods for measuring the system performance in terms of the switching ratio, the energy saving ratio, and the average response time of data frames, as well as giving the expressions for these performance measures. Numerical results are provided with analysis and simulation to show the average performance measures, standard deviations and the cost function with different system loads. Moreover, we construct a cost function with the aim of determining the optimal time length of the sleep window to minimize the cost function.

## First order rejection tests for multiple-objective optimization

### Journal of Global Optimization (2014-04-01) 58: 653-672 , April 01, 2014

Three rejection tests for multi-objective optimization problems based on first order optimality conditions are proposed. These tests can certify that a box does not contain any local minimizer, and thus it can be excluded from the search process. They generalize previously proposed rejection tests in several regards: Their scope include inequality and equality constrained smooth or nonsmooth multiple objective problems. Reported experiments show that they allow quite efficiently removing the cluster effect in mono-objective and multi-objective problems, which is one of the key issues in continuous global deterministic optimization.

## Invariant-point theorems and existence of solutions to optimization-related problems

### Journal of Global Optimization (2014-03-01) 58: 545-564 , March 01, 2014

To consider existence of solutions to various optimization-related problems, we first develop some equivalent versions of invariant-point theorems. Next, they are employed to derive sufficient conditions for the solution existence for two general models of variational relation and inclusion problems. We also prove the equivalence of these conditions with the above-mentioned invariant-point theorems. In applications, we include consequences of these results to a wide range of particular cases, from relatively general inclusion problems to classical results as Ekeland’s variational principle, and practical situations like traffic networks and non-cooperative games, to illustrate application possibilities of our general results. Many examples are provided to explain advantages of the obtained results and also to motivate in detail our problem settings.

## Stackelberg equilibria via variational inequalities and projections

### Journal of Global Optimization (2013-11-01) 57: 821-828 , November 01, 2013

Existence and location of Stackelberg equilibria is studied for two players by using appropriate variational inequalities and fixed point arguments. Both compact and non-compact strategy sets are considered in Euclidean spaces; in the non-compact case, we apply arguments from the theory of (discrete and continuous) projective dynamical systems. Some examples are also presented.

## Functional inequalities and theorems of the alternative involving composite functions

### Journal of Global Optimization (2014-08-01) 59: 837-863 , August 01, 2014

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form $$ f + g + k\circ H \ge h$$ . The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function $$ f + g + k\circ H$$ that are satisfied in various favorable circumstances. The results are then applied to the contexts of alternative type theorems involving composite and DC functions. The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations. As applications of these results, nonconvex optimization problems with composite functions, convex composite problems with conic constraints are examined at the end of the paper. There, strong duality, stable strong duality results for these classes of problems are established. Farkas-type results and stable form of these results for the corresponding systems involving composite functions are derived as well.