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Volume 1, 2011

Numerical Algebra, Control & Optimization

2011 , Volume 1 , Issue 3

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Preface
Shengji Li , Nan-Jing Huang and  Xinmin Yang
2011, 1(3): i-ii doi: 10.3934/naco.2011.1.3i +[Abstract](30) +[PDF](84.5KB)
Abstract:
This Special Issue of Numerical Algebra, Control and Optimization (NACO) is dedicated to Professor Franco Giannessi on the occasion of his 75th birthday and in recognition of his many fundamental contributions in Optimization and Nonlinear Analysis. It is a great honor and pleasure for the Guest Editors to have this opportunity to edit this Special Issue.

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General parametric sufficient optimality conditions for multiple objective fractional subset programming relating to generalized $(\rho,\eta,A)$ -invexity
Ram U. Verma
2011, 1(3): 333-339 doi: 10.3934/naco.2011.1.333 +[Abstract](39) +[PDF](59.2KB)
Abstract:
Motivated by the recent investigations, first a general framework for a class of $(\rho, \eta, A)$-invex n-set functions is introduced, and then some optimality conditions for multiple objective fractional programming on the generalized $(\rho, \eta, A)$-invexity are explored. The obtained results are general in nature and application-oriented to other investigations on fractional subset programming in literature.
Extragradient-projection method for solving constrained convex minimization problems
Luchuan Ceng , Qamrul Hasan Ansari and  Jen-Chih Yao
2011, 1(3): 341-359 doi: 10.3934/naco.2011.1.341 +[Abstract](39) +[PDF](252.4KB)
Abstract:
In this paper, we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a constrained convex minimization problem for a Fr\'{e}chet differentiable function. The iterative process is based on the so-called extragradient-projection method. We derive several weak convergence results for two sequences generated by the proposed iterative process. On the other hand, by applying the viscosity approximation method and the additional projection method (namely, the CQ method) to the extragradient-projection method, respectively, we also provide two modifications of the extragradient-projection method to obtain two strong convergence theorems. The results of this paper represent the supplement, improvement, extension and development of some known results given in the literature.
Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity
Xian-Jun Long and  Jing Quan
2011, 1(3): 361-370 doi: 10.3934/naco.2011.1.361 +[Abstract](43) +[PDF](162.1KB)
Abstract:
Using a parametric approach, we establish necessary and sufficient optimality conditions and derive some duality theorems for a class of nonsmooth minmax fractional programming problems containing generalized univex functions. The results obtained in this paper extend and improve some corresponding results in the literature.
Strong convergence theorems with three-step iteration in star-shaped metric spaces
Byung-Soo Lee
2011, 1(3): 371-379 doi: 10.3934/naco.2011.1.371 +[Abstract](30) +[PDF](162.5KB)
Abstract:
The author considers a Noor-type three-step iterative scheme including Ishikawa-type scheme as a special case to approximate common fixed points of an infinite family of uniformly quasi-Lipschitzian mappings and an infinite family of nonexpansive mappings in star-shaped metric spaces. His results are cases of star-shaped metric space of results shown in [7].
Some results on $l^k$-eigenvalues of tensor and related spectral radius
Chen Ling and  Liqun Qi
2011, 1(3): 381-388 doi: 10.3934/naco.2011.1.381 +[Abstract](42) +[PDF](160.0KB)
Abstract:
In this paper, we study the $l^k$-eigenvalues/vectors of a real symmetric square tensor. Specially, we investigate some properties on the related $l^k$-spectral radius of a real nonnegative symmetric square tensor.
Finding all minimal elements of a finite partially ordered set by genetic algorithm with a prescribed probability
Marcin Studniarski
2011, 1(3): 389-398 doi: 10.3934/naco.2011.1.389 +[Abstract](42) +[PDF](172.5KB)
Abstract:
For a general Markov chain model of genetic algorithm, we establish an upper bound for the number of iterations which must be executed in order to find, with a prescribed probability, an optimal solution in a finite multiobjective optimization problem.
An improved targeted climbing algorithm for linear programs
Mingfang Ding , Yanqun Liu and  John Anthony Gear
2011, 1(3): 399-405 doi: 10.3934/naco.2011.1.399 +[Abstract](44) +[PDF](133.0KB)
Abstract:
A class of exterior climbing algorithms (ladder algorithms) has been developed recently. Among this class, the targeted climbing algorithm has demonstrated very good numerical performance. In this paper an improved targeted climbing linear programming algorithm is proposed. A sequence of changing reference points, instead of a fixed reference point as in the original algorithm, is used in the improved algorithm to speed up the convergence. The improved algorithm is especially suited to solve problems involving many constraints close to the optimal solution---the case that causes many algorithms to slow down dramatically. Numerical tests and comparison with the simplex methods are presented. Our results show that the current algorithm works better on average than the simplex methods and is much faster for a large class of problems.
On linear vector optimization duality in infinite-dimensional spaces
Radu Ioan Boţ and  Sorin-Mihai Grad
2011, 1(3): 407-415 doi: 10.3934/naco.2011.1.407 +[Abstract](49) +[PDF](164.0KB)
Abstract:
In this paper we extend to infinite-dimensional spaces a vector duality concept recently considered in the literature in connection to the classical vector minimization linear optimization problem in a finite-dimensional framework. Weak, strong and converse duality for the vector dual problem introduced with this respect are proven and we also investigate its connections to some classical vector duals considered in the same framework in the literature.
Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization
Qilin Wang , Shengji Li and  Kok Lay Teo
2011, 1(3): 417-433 doi: 10.3934/naco.2011.1.417 +[Abstract](43) +[PDF](233.3KB)
Abstract:
In this paper, some properties are established for second-order adjacent derivatives of set-valued maps. Upper and lower semicontinuity and closedness are obtained for second-order adjacent derivatives of weak perturbation maps in vector optimization problems. Several examples are given for illustrating our results.
Orbital transfers: optimization methods and recent results
Mauro Pontani
2011, 1(3): 435-485 doi: 10.3934/naco.2011.1.435 +[Abstract](42) +[PDF](4130.0KB)
Abstract:
A wide variety of techniques have been employed in the past for optimizing orbital transfers, which represent the trajectories that lead a spacecraft from a given initial orbit to a specified final orbit. This paper describes several original approaches to optimizing impulsive and finite--thrust orbital transfers, and presents some very recent results. First, impulsive transfers between Keplerian trajectories are considered. A new, analytical optimization method applied to these transfers leads to conclusions of a global nature for transfers involving both ellipses and escape trajectories, without any limitation on the number of impulses, and with possible constraints on the radius of closest approach and greatest recession from the attracting body. A direct optimization technique, termed direct collocation with nonlinear programming algorithm, is then applied to finite--thrust transfers between circular orbits. Lastly, low--thrust orbital transfers are optimized through the joint use of the necessary conditions for optimality and of the recently introduced heuristic method referred to as particle swarm optimization. This work offers a complete description and demonstrates the effectiveness of the distinct techniques applied to optimizing orbital transfer problems of different nature.
A note on monotone approximations of minimum and maximum functions and multi-objective problems
Dušan M. Stipanović , Claire J. Tomlin and  George Leitmann
2011, 1(3): 487-493 doi: 10.3934/naco.2011.1.487 +[Abstract](35) +[PDF](154.8KB)
Abstract:
In paper [12] the problem of accomplishing multiple objectives by a number of agents represented as dynamic systems is considered. Each agent is assumed to have a goal which is to accomplish one or more objectives where each objective is mathematically formulated using an appropriate objective function. Sufficient conditions for accomplishing objectives are formulated using particular convergent approximations of minimum and maximum functions depending on the formulation of the goals and objectives. These approximations are differentiable functions and they monotonically converge to the corresponding minimum or maximum function. Finally, an illustrative pursuit-evasion game example of a capture of two evaders by two pursuers is provided.
    This note presents a preview of the treatment in [12].
A unified nonlinear augmented Lagrangian approach for nonconvex vector optimization
Chunrong Chen
2011, 1(3): 495-508 doi: 10.3934/naco.2011.1.495 +[Abstract](33) +[PDF](221.8KB)
Abstract:
In this paper, we propose a unified nonlinear augmented Lagrangian dual approach for a nonconvex vector optimization problem by applying a class of vector-valued nonlinear augmented Lagrangian penalty functions. We establish weak and strong duality results, necessary and sufficient conditions for uniformly exact penalization and exact penalization in the framework of nonlinear augmented Lagrangian.
A primal-dual algorithm for nonlinear programming exploiting negative curvature directions
Gianni Di Pillo , Giampaolo Liuzzi and  Stefano Lucidi
2011, 1(3): 509-528 doi: 10.3934/naco.2011.1.509 +[Abstract](42) +[PDF](292.1KB)
Abstract:
In this paper we propose a primal-dual algorithm for the solution of inequality constrained optimization problems. The distinguishing feature of the proposed algorithm is that of exploiting as much as possible the local non-convexity of the problem to the aim of producing a sequence of points converging to second order stationary points. In the unconstrained case this task is accomplished by computing a suitable negative curvature direction of the objective function. In the constrained case it is possible to gain analogous information by exploiting the non-convexity of a particular exact merit function. The algorithm hinges on the idea of comparing, at every iteration, the relative effects of two directions and then selecting the more promising one. The first direction conveys first order information on the problem and can be used to define a sequence of points converging toward a KKT pair of the problem. Whereas, the second direction conveys information on the local non-convexity of the problem and can be used to drive the algorithm away from regions of non-convexity. We propose a proper selection rule for these two directions which, under suitable assumptions, is able to generate a sequence of points that is globally convergent to KKT pairs that satisfy the second order necessary optimality conditions, with superlinear convergence rate if the KKT pair satisfies also the strong second order sufficiency optimality conditions.
A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem
Yafeng Li , Guo Sun and  Yiju Wang
2011, 1(3): 529-537 doi: 10.3934/naco.2011.1.529 +[Abstract](42) +[PDF](173.8KB)
Abstract:
For the polyhedral cone constrained eigenvalue problem over a polyhedral cone, based on its nonsmooth transformed version and a smoothing technique, we propose a modified smoothing Broyden-like method and establish its convergence under suitable conditions. The given computational experiments show the efficiency of the proposed method.
Asymptotic strong duality
Regina S. Burachik and  Xiaoqi Yang
2011, 1(3): 539-548 doi: 10.3934/naco.2011.1.539 +[Abstract](33) +[PDF](165.8KB)
Abstract:
Given a nonconvex and nonsmooth optimization problem, we define a family of ``perturbed'' Lagrangians, which induce well-behaved approximations of the dual problem. Our family of approximated problems is said to verify {\em strong asymptotic duality} when the optimal dual values of the perturbed problems approach the primal optimal value. Our perturbed Lagrangians can have the same order of smoothness as the functions of the original problem, a property not shared by the classical (unperturbed) augmented Lagrangian. Therefore our proposed scheme allows the use of efficient numerical methods for solving the perturbed dual problems. We establish general conditions under which strong asymptotic duality holds, and we relate the latter with both strong duality and lower semicontinuity of the perturbation function. We illustrate our perturbed duality scheme with two important examples: Constrained Nonsmooth Optimization and Nonlinear Semidefinite programming.
Bilevel mixed equilibrium problems in Banach spaces : existence and algorithmic aspects
Ouayl Chadli , Hicham Mahdioui and  Jen-Chih Yao
2011, 1(3): 549-561 doi: 10.3934/naco.2011.1.549 +[Abstract](39) +[PDF](183.0KB)
Abstract:
In this paper, we study the existence and algorithmic aspect of a class of bilevel mixed equilibrium problems (BMEP) in a Banach space setting. We introduce a suitable regularization of the bilevel mixed equilibrium problems by means of an auxiliary equilibrium problem. We show that it is possible to define from the auxiliary equilibrium problems a sequence strongly convergent to a solution of the bilevel mixed equilibrium problem. The results obtained are interesting in the sense that they improve some new results on bilevel mixed equilibrium problems and give answer to some open questions in literature related to the convergence of algorithms for (BMEP).

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