Showing papers in "Optimization Letters in 2014"
TL;DR: It is proved that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping which is the unique solution of the variational inequality problem.
Abstract: In this paper, we introduce and study an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping in real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping which is the unique solution of the variational inequality problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area.
99 citations
TL;DR: An iterative method for solving absolute value equations gives a sufficient condition for unique solvability of these equations for arbitrary right-hand sides.
Abstract: We describe an iterative method for solving absolute value equations. The result gives a sufficient condition for unique solvability of these equations for arbitrary right-hand sides. This sufficient condition is compared with that one by Mangasarian and Meyer.
86 citations
TL;DR: The error analysis is presented to prove the convergence order of nonlinear systems using iterative methods and a general class of multi-point iteration methods with various orders is constructed.
Abstract: In this article, the numerical solution of nonlinear systems using iterative methods are dealt with. Toward this goal, a general class of multi-point iteration methods with various orders is constructed. The error analysis is presented to prove the convergence order. Also, a thorough discussion on the computational complexity of the new iterative methods will be given. The analytical discussion of the paper will finally be upheld through solving some application-oriented problems.
81 citations
TL;DR: The sufficient conditions for the convergence of the Picard–HSS iteration method for AVE are given and some numerical experiments are given to show the effectiveness of the method and to compare with two available methods.
Abstract: Recently Bai and Yang in (Appl Numer Math 59:2923–2936, 2009) proposed the Picard–Hermitian and skew-Hermitian splitting (HSS) iteration method to solve the system of nonlinear equations \(Ax=\varphi (x)\), where \(A\in \mathbb {C}^{n \times n}\) is a non-Hermitian positive definite matrix and \(\varphi :\mathbb {D}\subset \mathbb {C}^n \rightarrow \mathbb {C}^n\) is continuously differentiable function defined on the open complex domain \(\mathbb {D}\) in the \(n\)-dimensional complex linear space \(\mathbb {C}^n\). In this paper, we focus our attention to the absolute value equation (AVE) \(Ax=\varphi (x)\) where \(\varphi (x)=|x|+b\), where \(b\in \mathbb {C}^n\). Since the function \(\varphi \) in AVE is not continuously differentiable function the convergence analysis of the Picard–HSS iteration method for this problem needs to be investigated. We give sufficient conditions for the convergence of the Picard–HSS iteration method for AVE. Some numerical experiments are given to show the effectiveness of the method and to compare with two available methods.
80 citations
TL;DR: In this paper, a split proximal algorithm with a way of selecting the step-sizes such that its implementation does not need any prior information about the operator norm is proposed.
Abstract: In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function \(f\) such that its image under a bounded linear operator \(A\) minimizes another convex function \(g\). Based on an idea introduced in Lopez (Inverse Probl 28:085004, 2012), we propose a split proximal algorithm with a way of selecting the step-sizes such that its implementation does not need any prior information about the operator norm. Because the calculation or at least an estimate of the operator norm \(\Vert A\Vert \) is not an easy task. Secondly, we investigate the case where one of the two involved functions is prox-regular, the novelty of this approach is that the associated proximal mapping is not nonexpansive any longer. Such situation is encountered, for instance, in numerical solution to phase retrieval problem in crystallography, astronomy and inverse scattering Luke (SIAM Rev 44:169–224, 2002) and is therefore of great practical interest.
79 citations
TL;DR: This work develops more compact linear 0–1 formulations for the considered types of problems with $$\varTheta (n^2)$$Θ(n2) entities and provides reformulations and valid inequalities that improve the performance of the developed models.
Abstract: Critical node detection problems aim to optimally delete a subset of nodes in order to optimize or restrict a certain metric of network fragmentation. In this paper, we consider two network disruption metrics which have recently received substantial attention in the literature: the size of the remaining connected components and the total number of node pairs connected by a path. Exact solution methods known to date are based on linear 0–1 formulations with at least $$\varTheta (n^3)$$
entities and allow one to solve these problems to optimality only in small sparse networks with up to 150 nodes. In this work, we develop more compact linear 0–1 formulations for the considered types of problems with $$\varTheta (n^2)$$
entities. We also provide reformulations and valid inequalities that improve the performance of the developed models. Computational experiments show that the proposed formulations allow finding exact solutions to the considered problems for real-world sparse networks up to 10 times larger and with CPU time up to 1,000 times faster compared to previous studies.
77 citations
TL;DR: The Karush–Kuhn–Tucker optimality conditions are derived for LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuous differentiablity and Hukuhara differentiability.
Abstract: In this paper, we study the Karush–Kuhn–Tucker optimality conditions in a class of nonconvex optimization problems with an interval-valued objective function. Firstly, the concepts of preinvexity and invexity are extended to interval-valued functions. Secondly, several properties of interval-valued preinvex and invex functions are investigated. Thirdly, the KKT optimality conditions are derived for LU-preinvex and invex optimization problems with an interval-valued objective function under the conditions of weakly continuous differentiablity and Hukuhara differentiablity. Finally, the relationships between a class of variational-like inequalities and the interval-valued optimization problems are established.
58 citations
TL;DR: By imposing assumptions of generalized convexity, this paper gives sufficient conditions for efficient solutions for nonsmooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite.
Abstract: This paper is devoted to the study of nonsmooth multiobjective semi-infinite programming problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these problems, and then necessary optimality conditions for weakly efficient solutions are investigated. Finally by imposing assumptions of generalized convexity we give sufficient conditions for efficient solutions.
56 citations
TL;DR: This paper proposes a method for testing basis stability and even though the method is exponential in the worst case (not surprisingly due to NP-hardness of the problem), it is fast in many cases.
Abstract: Interval linear programming (ILP) was introduced in order to deal with linear programming problems with uncertainties that are modelled by ranges of admissible values. Basic tasks in ILP such as calculating the optimal value bounds or set of all possible solutions may be computationally very expensive. However, if some basis stability criterion holds true then the problems becomes much more easy to solve. In this paper, we propose a method for testing basis stability. Even though the method is exponential in the worst case (not surprisingly due to NP-hardness of the problem), it is fast in many cases.
55 citations
TL;DR: A genetic algorithm called GENALGO is presented to solve large single row facility layout problem instances and improves the previously best known solutions for the 19 instances of 58 benchmark instances and is competitive for most of the remaining ones.
Abstract: The single row facility layout is the NP-Hard problem of arranging facilities with given lengths on a line, so as to minimize the weighted sum of the distances between all pairs of facilities. Owing to its computational complexity, researchers have developed several heuristics to obtain good quality solutions. In this paper, we present a genetic algorithm called GENALGO to solve large single row facility layout problem instances. Our algorithm uses standard genetic operators and periodically improves the fitness of all individuals. Our computational experiments show that our genetic algorithm yields high quality solutions in spite of starting with an initial population that is randomly generated. Our algorithm improves the previously best known solutions for the 19 instances of 58 benchmark instances and is competitive for most of the remaining ones.
51 citations
TL;DR: This paper provides a polynomial-time algorithm to find the optimal job sequence,Due date values, and resource allocations that minimize an integrated objective function, which includes earliness, tardiness, due date assignment, and total resource consumption costs.
Abstract: In this paper, we consider a single-machine earliness-tardiness scheduling problem with due-date assignment, in which the processing time of a job is a function of its position in a sequence and its resource allocation. The due date assignment methods studied include the common due date, and the slack due date, which reflects equal waiting time allowance for the jobs. For each combination of due date assignment method and processing time function, we provide a polynomial-time algorithm to find the optimal job sequence, due date values, and resource allocations that minimize an integrated objective function, which includes earliness, tardiness, due date assignment, and total resource consumption costs.
TL;DR: A MILP model for an extended version of the Flexible Job Shop Scheduling problem is proposed, which allows the precedences between operations of a job to be given by an arbitrary directed acyclic graph rather than a linear order.
Abstract: A MILP model for an extended version of the Flexible Job Shop Scheduling problem is proposed. The extension allows the precedences between operations of a job to be given by an arbitrary directed acyclic graph rather than a linear order. The goal is the minimization of the makespan. Theoretical and practical advantages of the proposed model are discussed. Numerical experiments show the performance of a commercial exact solver when applied to the proposed model. The new model is also compared with a simple extension of the model described by Ozguven et al. (Appl Math Modell 34:1539–1548, 2010), using instances from the literature and instances inspired by real data from the printing industry.
TL;DR: This paper proposes a general scheduled service network design modelling framework that captures the fundamental concepts related to the definition of urban-vehicle tactical plans within a two-tier distribution network.
Abstract: This paper focuses on two-tier city logistics systems for advanced management of urban freight activities and, in particular, on the first layer of such systems where freight is moved from distribution centers on the outskirts of the city to satellite platforms by urban vehicles, from where it will be distributed to customers by a different fleet of dedicated vehicles. We address the issue of planning the services of this first tier system, that is, select services, their routes and schedules, and determine the itineraries of the customer-demand flows through these facilities and services. We propose a general scheduled service network design modelling framework that captures the fundamental concepts related to the definition of urban-vehicle tactical plans within a two-tier distribution network. We examine several operational assumptions regarding the management of the urban-vehicle fleet and the flexibility associated with the delivery of goods, and show how the proposed modelling framework can evolve to represent an increasing level of detail. A discussion of algorithmic perspectives completes the paper.
TL;DR: It is shown that a version of the proposed scheme leads to a tractable convex relaxation when the chance constraint function is affine with respect to the underlying random vector and the random vector has independent components.
Abstract: In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed scheme leads to a tractable convex relaxation when the chance constraint function is affine with respect to the underlying random vector and the random vector has independent components. We also propose an iterative improvement scheme for refining the bounds.
TL;DR: The proposed LQ control model for stochastic singular systems provides an appropriate and effective framework to study the portfolio selection problem in light of the recent development on general stochastics LQ problems.
Abstract: In this paper, problems of stability and optimal control for a class of stochastic singular systems are studied. Firstly, under some appropriate assumptions, some new results about mean-square admissibility are developed and the corresponding LMI sufficient condition is given. Secondly, finite-time horizon and infinite-time horizon linear quadratic (LQ) control problems for the stochastic singular system are investigated, in which the coefficients are allowed to be random in control input and quadratic criterion. Some results involving new stochastic generalized Riccati equation are discussed as well. Finally, the proposed LQ control model for stochastic singular systems provides an appropriate and effective framework to study the portfolio selection problem in light of the recent development on general stochastic LQ problems.
TL;DR: The new family of CVaR norms are piece-wise linear functions on Rn and can be used in various applications where the Euclidean norm is typically used, and allows formulating this problem as a convex or linear program for any level of conservativeness.
Abstract: This paper introduces the family of CVaR norms in $${\mathbb {R}}^{n}$$
, based on the CVaR concept. The CVaR norm is defined in two variations: scaled and non-scaled. The well-known $$L_{1}$$
and $$L_{\infty }$$
norms are limiting cases of the new family of norms. The D-norm, used in robust optimization, is equivalent to the non-scaled CVaR norm. We present two relatively simple definitions of the CVaR norm: (i) as the average or the sum of some percentage of largest absolute values of components of vector; (ii) as an optimal solution of a CVaR minimization problem suggested by Rockafellar and Uryasev. CVaR norms are piece-wise linear functions on $${\mathbb {R}}^{n}$$
and can be used in various applications where the Euclidean norm is typically used. To illustrate, in the computational experiments we consider the problem of projecting a point onto a polyhedral set. The CVaR norm allows formulating this problem as a convex or linear program for any level of conservativeness.
TL;DR: It is proved that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, i.e., they are slices of the cone of positive semidefinite matrices, using the matrix-tree theorem.
Abstract: We prove that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, i.e., they are slices of the cone of positive semidefinite matrices. The proof uses the matrix-tree theorem, an idea already present in Choe et al.
TL;DR: In this article, a column generation approach embedded within a branch-and-price (B&P) procedure was proposed to find a schedule that minimizes the completion time (makespan) of a project, composed of a set of activities.
Abstract: This work introduces a procedure to solve the multi-skill project scheduling problem (MSPSP) (Neron and Baptista, International symposium on combinatorial, optimization (CO’2002), 2002). The MSPSP mixes both the classical resource constrained project scheduling problem and the multi-purpose machine model. The aim is to find a schedule that minimizes the completion time (makespan) of a project, composed of a set of activities. In addition, precedence relations and resources constraints are considered. In this problem, resources are staff members that master several skills. Thus, a given number of workers must be assigned to perform each skill required by an activity. Practical applications include the construction of buildings, as well as production and software development planning. We present a column generation approach embedded within a branch-and-price (B&P) procedure that considers a given activity and time-based decomposition approach. Obtained results show that the proposed B&P procedure is able to reach optimal solutions for several small and medium sized instances in an acceptable computational time. Furthermore, some previously open instances were optimally solved.
TL;DR: In this article, the coloring problem for graph classes defined by two small forbidden induced subgraphs is studied and sufficient conditions for effective solvability of the problem in such classes are proved.
Abstract: The coloring problem is studied in the paper for graph classes defined by two small forbidden induced subgraphs. We prove some sufficient conditions for effective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most five vertices except 13 explicitly enumerated cases.
TL;DR: A concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two by solving 2 or less linear programs per LCP problem.
Abstract: We consider the linear complementarity problem (LCP): $$Mz+q\ge 0, z\ge 0, z^{\prime }(Mz+q)=0$$
as an absolute value equation (AVE): $$(M+I)z+q=|(M-I)z+q|$$
, where $$M$$
is an $$n\times n$$
square matrix and $$I$$
is the identity matrix. We propose a concave minimization algorithm for solving (AVE) that consists of solving a few linear programs, typically two. The algorithm was tested on 500 consecutively generated random solvable instances of the LCP with $$n=10, 50, 100, 500$$
and 1,000. The algorithm solved $$100\,\%$$
of the test problems to an accuracy of $$10^{-8}$$
by solving 2 or less linear programs per LCP problem.
TL;DR: This work considers an extention where supply is also random so that the quantity ordered is not necessarily received in full at the beginning of the period and focuses on the resulting optimization problem and provides interesting characterizations on the optimal order quantity.
Abstract: The newsvendor model is perhaps the most widely analyzed model in inventory management. In this single-period model, the only source of randomness is the demand during the period and one tries to determine the optimal order quantity in view of various cost factors. We consider an extention where supply is also random so that the quantity ordered is not necessarily received in full at the beginning of the period. Such models have been well-received in the literature with the assumption of independence between demand and supply. In this setting, we suppose that the random demand and supply are not necessarily independent. We focus on the resulting optimization problem and provide interesting characterizations on the optimal order quantity.
TL;DR: The upper, lower semicontinuity and closedness of the minimal solution and weak minimal solution set mappings to a parametric set-valued vector optimization problem with set optimization criterion under some suitable assumptions are established.
Abstract: In this paper, we establish the upper, lower semicontinuity and closedness of the minimal solution and weak minimal solution set mappings to a parametric set-valued vector optimization problem with set optimization criterion under some suitable assumptions.
TL;DR: The correntropic loss function is presented, which is a smooth and robust measure that can be utilized in optimization based data analysis methods.
Abstract: Similarity measures play a critical role in the solution quality of data analysis methods. Outliers or noise often taint the solution, hence, practical data analysis calls for robust measures. The correntropic loss function is a smooth and robust measure. In this paper, we present the properties of the correntropic loss function that can be utilized in optimization based data analysis methods.
TL;DR: This paper considers the single machine scheduling problem with truncated job-dependent learning effect and several polynomial time algorithms are proposed to optimally solve the problems with the above objective functions.
Abstract: In this paper we consider the single machine scheduling problem with truncated job-dependent learning effect. By the truncated job-dependent learning effect, we mean that the actual job processing time is a function which depends not only on the job-dependent learning effect (i.e., the learning in the production process of some jobs to be faster than that of others) but also on a control parameter. The objectives are to minimize the makespan, the total completion time, the total absolute deviation of completion time, the earliness, tardiness and common (slack) due-date penalty, respectively. Several polynomial time algorithms are proposed to optimally solve the problems with the above objective functions.
TL;DR: In this paper parallel identical machines scheduling problems with deteriorating jobs and learning effects are considered and it is shown that the problems remain polynomially solvable under the proposed model.
Abstract: In this paper parallel identical machines scheduling problems with deteriorating jobs and learning effects are considered. In this model, job processing times are defined by functions of their starting times and positions in the sequence. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time and total absolute differences in completion times; minimizing a cost function containing total waiting time and total absolute differences in waiting times. We show that the problems remain polynomially solvable under the proposed model.
TL;DR: A mathematical model of the problem of packing unequal circles into a rectangular strip with fixed width and minimal length is constructed and the idea of increasing the dimension of the solution space by assuming radii of circles to be variables is developed.
Abstract: The paper considers a problem of packing unequal circles into a rectangular strip with fixed width and minimal length. We develop the idea of increasing the dimension of the solution space by assuming radii of circles to be variables. A mathematical model of the problem is constructed and its characteristics are investigated. Taking into account the characteristics we offer a solution strategy of the problem including a number of non-linear programming subproblems of packing circles of variable radii. The solution strategy involves special ways of construction of starting points, calculation of local minima, jump from one local extremum to another, decrease of the problem dimension and rearrangement of pairs of circles. For calculating local extrema an interior point optimizer together with the concept of active inequalities are used. We compare 146 numerical benchmark examples and give seven new ones for 125, 150, 175, 225, 250, 275 and 300 circles.
TL;DR: Lower and upper bound approximation for the best case optimal value, and suitable methods for both of them are proposed, and a not apriori exponential algorithm for computing the best cases optimal value is proposed.
Abstract: Interval linear programming addresses problems with uncertain coefficients and the only information that we have is that the true values lie somewhere in the prescribed intervals. For the inequality constraint problem, computing the worst case scenario and the corresponding optimal value is an easy task, but the best case optimal value calculation is known to be NP-hard. In this paper, we discuss lower and upper bound approximation for the best case optimal value, and propose suitable methods for both of them. We also propose a not apriori exponential algorithm for computing the best case optimal value. The presented techniques are tested by randomly generated data, and also applied in a simple data classification problem.
TL;DR: Under the proposed model, the setup time is past-sequence-dependent and the actual job processing time is a general function of the processing times of the jobs already processed and its scheduled position.
Abstract: Scheduling with learning effect and deteriorating jobs has become more popular. However, most of the research assume that the setup time is negligible or a part of the job processing time. In this paper, we propose a model where the deteriorating jobs, the learning effect, and the setup times are present simultaneously. Under the proposed model, the setup time is past-sequence-dependent and the actual job processing time is a general function of the processing times of the jobs already processed and its scheduled position. We provide the optimal schedules for some single-machine problems.
TL;DR: A strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space is proposed.
Abstract: We propose a strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions.
TL;DR: A modified nonmonotone BFGS algorithm is developed for solving a smooth system of nonlinear equations using an algorithmic parameter controlling the magnitude of nonmonotonicity such that the numerical performance of the developed algorithm is improved.
Abstract: In this paper, a modified nonmonotone BFGS algorithm is developed for solving a smooth system of nonlinear equations. Different from the existent techniques of nonmonotone line search, the value of an algorithmic parameter controlling the magnitude of nonmonotonicity is updated at each iteration by the known information of the system of nonlinear equations such that the numerical performance of the developed algorithm is improved. Under some suitable assumptions, the global convergence of the algorithm is established for solving a generic nonlinear system of equations. Implementing the algorithm to solve some benchmark test problems, the obtained numerical results demonstrate that it is more effective than some similar algorithms available in the literature.