We deal with complementarity problems over second-order cones. The complementarity problem is an important class of problems in the real world and involves many optimization problems. The complementarity problem can be reformulated as a nonsmooth system of equations. Based on the smoothed Fischer-Burmeister function, we construct a smoothing Newton method for solving such a nonsmooth system. The proposed method controls a smoothing parameter appropriately. We show the global and quadratic convergence of the method. Finally, some numerical results are given.

We investigate the optimal scheduling of retrieval jobs for double-deep type Automated Storage and Retrieval Systems (AS/RS) in the Flexible Manufacturing System (FMS) used in modern industrial production. Three types of evolutionary algorithms, the Genetic Algorithm (GA), the Immune Genetic Algorithm (IGA), and the Particle Swarm Optimization (PSO) algorithm, are implemented to obtain the optimal assignments. The objective is to minimize the working distance, that is, the shortest retrieval time travelled by the Storage and Retrieval (S/R) machine. Simulation results and comparisons show the advantages and feasibility of the proposed methods.

Recently, Gandomi and Alavi proposed a novel swarm intelligent method, called krill herd (KH), for global optimization. To enhance the performance of the KH method, in this paper, a new improved meta-heuristic simulated annealing-based krill herd (SKH) method is proposed for optimization tasks. A new krill selecting (KS) operator is used to refine krill behavior when updating krill’s position so as to enhance its reliability and robustness dealing with optimization problems. The introduced KS operator involves greedy strategy and accepting few not-so-good solutions with a low probability originally used in simulated annealing (SA). In addition, a kind of elitism scheme is used to save the best individuals in the population in the process of the krill updating. The merits of these improvements are verified by fourteen standard benchmarking functions and experimental results show that, in most cases, the performance of this improved meta-heuristic SKH method is superior to, or at least highly competitive with, the standard KH and other optimization methods.

An implementable algorithm for solving a nonsmooth convex optimization problem is proposed by combining Moreau-Yosida regularization and bundle and quasi-Newton ideas. In contrast with quasi-Newton bundle methods of Mifflin et al. (1998), we only assume that the values of the objective function and its subgradients are evaluated approximately, which makes the method easier to implement. Under some reasonable assumptions, the proposed method is shown to have a Q-superlinear rate of convergence.

The nonfragile robust finite-time $L_2$-$L_{\infty }$ control problem for a class of nonlinear uncertain systems with uncertainties and time-delays is considered. The nonlinear parameters are considered to satisfy the Lipschitz conditions and the exogenous disturbances are unknown but energy bounded. By using the Lyapunov function approach, the sufficient condition for the existence of nonfragile robust finite-time $L_2$-$L_{\infty }$ controller is given in terms of linear matrix inequalities (LMIs). The finite-time controller is designed such that the resulting closed-loop system is finite-time bounded for all admissible uncertainties and satisfies the given $L_2$-$L_{\infty }$ control index. Simulation results illustrate the validity of the proposed approach.

Low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, and system identification and control. This class of optimization problems is generally $𝒩𝒫$ hard. A popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend and characterize the concept of $s$-goodness for a sensing matrix in sparse signal recovery (proposed by Juditsky and Nemirovski (Math Program, 2011)) to linear transformations in LMR. Using the two characteristic $s$-goodness constants, ${\gamma }_s$ and ${\widehat{\gamma }}_s$, of a linear transformation, we derive necessary and sufficient conditions for a linear transformation to be $s$-good. Moreover, we establish the equivalence of $s$-goodness and the null space properties. Therefore, $s$-goodness is a necessary and sufficient condition for exact $s$-rank matrix recovery via the nuclear norm minimization.

We establish a new second-order iteration method for solving nonlinear equations. The efficiency index of the method is 1.4142 which is the same as the Newton-Raphson method. By using some examples, the efficiency of the method is also discussed. It is worth to note that (i) our method is performing very well in comparison to the fixed point method and the method discussed in Babolian and Biazar (2002) and (ii) our method is so simple to apply in comparison to the method discussed in Babolian and Biazar (2002) and involves only first-order derivative but showing second-order convergence and this is not the case in Babolian and Biazar (2002), where the method requires the computations of higher-order derivatives of the nonlinear operator involved in the functional equation.

We obtain a new Taylor's formula in terms of the $k+1$ order subdifferential of a $C^{k,1}$ function from $R^n$ to $R^m$. As its applications in optimization problems, we build $k+1$ order sufficient optimality conditions of this kind of functions and $k+1$ order necessary conditions for strongly $C$-quasiconvex functions.

This paper investigates the set-valued complementarity problems (SVCP) which poses rather different features from those that classical complementarity problems hold, due to tthe fact that he index set is not fixed, but dependent on $x$. While comparing the set-valued complementarity problems with the classical complementarity problems, we analyze the solution set of SVCP. Moreover, properties of merit functions for SVCP are studied, such being as level bounded and error bounded. Finally, some possible research directions are discussed.