Abstract: We suggest an iterative method for solving absolute value equation Ax − |x| = b, where \({A\in R^{n\times n}}\) is symmetric matrix and \({b\in R^{n}}\), coupled with the minimization technique. We also discuss the convergence of the proposed method. Some examples are given to illustrate the implementation and efficiency of the method.

Abstract: Given a weighted, undirected simple graph G = (V, E, d) (where \({d:E\to\mathbb{R}_+}\)), the distance geometry problem (DGP) is to determine an embedding \({x:V\to\mathbb{R}^K}\) such that \({\forall \{i,j\} \in E\;\|x_i-x_j\|=d_{ij}}\) . Although, in general, the DGP is solved using continuous methods, under certain conditions the search is reduced to a discrete set of points. We give one such condition as a particular order on V. We formalize the decision problem of determining whether such an order exists for a given graph and show that this problem is NP-complete in general and polynomial for fixed dimension K. We present results of computational experiments on a set of protein backbones whose natural atomic order does not satisfy the order requirements and compare our approach with some available continuous space searches.

Abstract: We consider a biodiesel production company that collects waste vegetable oil from source points that generate waste in large amounts. The company uses the collected waste as raw material for biodiesel production. The manager of this company needs to decide which of the present source points to include in the collection program, which of them to visit on each day, which periodic routing schedule to repeat over an infinite horizon and how many vehicles to operate such that the total collection, inventory and purchasing costs are minimized while the production requirements and operational constraints are met. For this selective and periodic inventory routing problem, we propose two different formulations, compare them and apply the better performing one on a real-world problem with 36 scenarios. We generate lower bounds using a partial linear relaxation model, and observe that the solutions obtained through our model are within 3.28% of optimality on the average. Several insights regarding the customer selection, routing and purchasing decisions are acquired with sensitivity analysis.

Abstract: We propose a simple privacy-preserving reformulation of a linear program whose equality constraint matrix is partitioned into groups of rows. Each group of matrix rows and its corresponding right hand side vector are owned by a distinct private entity that is unwilling to share or make public its row group or right hand side vector. By multiplying each privately held constraint group by an appropriately generated and privately held random matrix, the original linear program is transformed into an equivalent one that does not reveal any of the privately held data or make it public. The solution vector of the transformed secure linear program is publicly generated and is available to all entities.

Abstract: In this paper, we consider the single-machine scheduling problems with nonlinear deterioration. By the nonlinear deterioration effect, we mean that the processing times of jobs are nonlinear functions of their starting times. We show that even with the introduction of nonlinear deterioration to job processing times, single machine makespan minimization problem remains polynomially solvable. We also show that an optimal schedule of the total completion time minimization problem is V-shaped with respect to job normal processing times. A heuristic algorithm utilizing the V-shaped property is proposed, and computational experiments show that it performs effectively and efficiently in obtaining near-optimal solutions.

Abstract: Bees Algorithm is one of the swarm intelligence based heuristics which tries to model natural behaviour of honey bees in food foraging and used to solve optimization problems. On the other hand, Two-sided Assembly Line Balancing Problem is a generalization of simple Assembly Line Balancing Problem where different assembly tasks are carried out on the same product in parallel at both left and right sides of the line. Two-sided assembly lines are generally employed for the assembly of large-sized products such as buses and trucks. Furthermore, many real life problems contain imprecise objectives and Fuzzy Multi-objective Programming gives an opportunity to handle such situations. In this study, Two-sided Assembly Line Balancing Problem is considered more realistically by employing positional, zoning and synchronous task constraints and by utilizing fuzzy approaches so as to maximize work slackness index and line efficiency, and minimize total balance delay. For solving this problem Bees Algorithm is used as a search mechanism for obtaining good solutions and extensive computational results are presented.

Abstract: We consider a two-machine flow shop scheduling problem with effects of deterioration and learning. By the effects of deterioration and learning, we mean that the processing time of a job is a function of its execution starting time and its position in a sequence. The objective is to find a sequence that minimizes the makespan. Several dominance properties and two lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm proposed to solve the problem. Two heuristic algorithms are also proposed to obtain near-optimal solutions. Computational results are presented to evaluate the performance of the proposed algorithms.

Abstract: Abkar and Gabeleh in (J. Optim. Theory. Appl. doi: 10.1007/s10957-011-9818-2) proved some theorems which ensure the existence and convergence of fixed points, as well as best proximity points for cyclic mappings in ordered metric spaces. In this paper we extend these results to generalized cyclic contractions and obtain some new results on the existence and convergence of fixed points for weakly contractive mappings, as well as on best proximity points for cyclic φ-contraction mappings in partially ordered metric spaces.

Abstract: Very recently, Dang and Gao (Inverse Probl 27:015007, 2011) introduced a KM-CQ algorithm with strong convergence for the split feasibility problem. In this paper, we will continue to consider the split feasibility problem. We present two algorithms. First, we introduce an implicit algorithm. Consequently, by discretizing the continuous implicit algorithm, we obtain an explicit algorithm. Under some weaker conditions, we show the strong convergence of presented algorithms to some solution of the split feasibility problem which solves some special variational inequality. As special cases, we obtain two algorithms which converge strongly to the minimum norm solution of the split feasibility problem. Results obtained in this paper include the corresponding results of Dang and Gao (2011) and extend a recent result of Wang and Xu (J Inequalities Appl 2010, doi:10.1155/2010/102085).

Abstract: We present the first approximation algorithm for a two depot, heterogeneous traveling salesman problem with an approximation ratio of 3 when the costs are symmetric and satisfy the triangle inequality.

Abstract: A nonlinear multiobjective optimization problem is considered. Two methods are proposed to generate solutions with an approximately uniform distribution in a Pareto set. The first method is supposed to find the solutions as minimizers of weighted sums of objective functions where the weights are properly selected using a branch and bound type algorithm. The second method is based on lexicographic goal programming. The proposed methods are compared with several metaheuristic methods using two and three-criteria tests and applied problems.

Abstract: We present a biased random-key genetic algorithm (BRKGA) for finding small covers of computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. Using a parallel implementation of the BRKGA, we compute improved covers for the two largest instances in a standard set of test problems used to evaluate solution procedures for this problem. The new covers for instances A 405 and A 729 have sizes 335 and 617, respectively. On all other smaller instances our algorithm consistently produces covers of optimal size.

Abstract: In this paper, we consider a task allocation model that consists of assigning a set of m unmanned aerial vehicles (UAVs) to a set of n tasks in an optimal way. The optimality is quantified by target scores. The mission is to maximize the target score while satisfying capacity constraints of both the UAVs and the tasks. This problem is known to be NP-hard. Existing algorithms are not suitable for the large scale setting. Scalability and robustness are recognized as two main issues. We deal with these issues by two optimization approaches. The first approach is the Cross-Entropy (CE) method, a generic and practical tool of stochastic optimization for solving NP-hard problem. The second one is Branch and Bound algorithm, an efficient classical tool of global deterministic optimization. The numerical results show the efficiency of our approaches, in particular the CE method for very large scale setting.

Abstract: We provide an explicit description of the convex hull of the set of matrices of bounded rank, restricted to balls for the spectral norm. As applications, we deduce two relaxed forms of the rank function restricted to balls for the spectral norm: one is the quasiconvex hull of this rank function, another one is the convex hull of the rank function, thus retrieving Fazel’s theorem (Matrix rank minimization with applications, 2002).

Abstract: The shortest path tour problem (SPTP) consists in finding a shortest path from a given origination node s to a given destination node d in a directed graph with nonnegative arc lengths with the constraint that the optimal path P should successively and sequentially pass through at least one node from given node subsets T1, T2, . . . , TN, where \({T_i \cap T_j = \emptyset, \forall\ i, j=1,\ldots,N,\ i eq j}\). In this paper, it will proved that the SPTP belongs to the complexity class P and several alternative techniques will be presented to solve it.

Abstract: Conjugate gradient methods have been widely used as schemes to solve large-scale unconstrained optimization problems. The search directions for the conventional methods are defined by using the gradient of the objective function. This paper proposes two nonlinear conjugate gradient methods which take into account mostly information about the objective function. We prove that they converge globally and numerically compare them with conventional methods. The results show that with slight modification to the direction, one of our methods performs as well as the best conventional method employing the Hestenes–Stiefel formula.

Abstract: In this paper, we consider the design problem of a public service facility network with existing facilities when there is a threat of possible terrorist attacks. The aim of the system planner, who is responsible for the operation of the network, is to open new facilities, relocate existing ones if necessary, and protect some of the facilities to ensure a maximum coverage of the demand that is assumed to be aggregated at customer zones. By doing so, the system planner anticipates that a number of unprotected facilities will be rendered out-of-service by terrorist attacks. It is assumed that the sum of the fixed cost of opening new facilities, the relocation costs, and the protection costs cannot exceed a predetermined budget level. Adopting the approach of gradual (or partial) coverage, we formulate a bilevel programming model where the system planner is the leader and the attacker is the follower. The objective of the former is the maximization of the total service coverage, whereas the latter wants to minimize it. We propose a heuristic solution procedure based on tabu search where the search space consists of the decisions of the system planner, and the corresponding objective value is computed by optimally solving the attacker’s problem using CPLEX. To assess the quality of the solutions produced by the tabu search (TS) heuristic, we also develop an exhaustive enumeration method, which explores all the possible combinations of opening new facilities, relocating existing ones, and protecting them. Since its time complexity is exponential, it can only be used for relatively small instances. Therefore, to be used as a benchmark method, we also implement a hill climbing procedure employed with the same type of moves as the TS heuristic. Besides, we carry out a sensitivity analysis on some of the problem parameters to investigate their effect on the solution characteristics.

Abstract: This article presents a new 2-approximation algorithm for a multiple depot, multiple terminal, Hamiltonian path problem when the costs satisfy the triangle inequality. For the case where all the salesmen start from the same depot, we present another algorithm with an approximation ratio of $${\frac{5}{3}}$$ . These results generalize the approximation algorithms currently available for the single depot, single terminal Hamiltonian path problem.

Abstract: Strongly motivated by its possible applications in Mechanics, in our previous work (Pitea and Postolache (Optim. Lett. doi:10.1007/s11590-010-0272-0, 2011)), we initiated an optimization theory for the second order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multi-time multi-objective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary conditions. As natural continuation of our results in Pitea and Postolache (Optim. Lett. doi:10.1007/s11590-010-0272-0, 2011), the present work introduces a study of sufficient efficiency conditions. While the background in Sect. 2 is introductory, the theory in Sect. 3 is new as a whole, containing our results.

Abstract: This paper concerns developing two hybrid proximal point methods (PPMs) for finding a common solution of some optimization-related problems. First we construct an algorithm to solve simultaneously an equilibrium problem and a variational inequality problem, combing the extragradient method for variational inequalities with an approximate PPM for equilibrium problems. Next we develop another algorithm based on an alternate approximate PPM for finding a common solution of two different equilibrium problems. We prove the global convergence of both algorithms under pseudomonotonicity assumptions.

Abstract: The recent research on linearization techniques for solving 0-1 quadratic programming problems focuses on providing concise models and tightening constraint bounds. In this paper, we propose a computational enhancement for a linearization technique to make the linearized model much faster to solve. We investigate the computational performance of the proposed approach, by comparing it with other linearization techniques on a class of 0-1 quadratic programming problems. We can further speed up the proposed technique by heuristically tightening the constraint bounds, as demonstrated by solving the uncapacitated single allocation p-hub median problem using the Civil Aeronautics Board data.

Abstract: In this paper, the Minty vector variational-like inequality, the Stampacchia vector variational-like inequality, and the weak formulations of these two inequalities defined by means of Mordukhovich limiting subdifferentials are introduced and studied in Asplund spaces. Some relations between the vector variational-like inequalities and vector optimization problems are established by using the properties of Mordukhovich limiting subdifferentials. An existence theorem of solutions for the weak Minty vector variational-like inequality is also given.

Abstract: We consider single-machine scheduling and slack due-date assignment problems simultaneously with the position-dependent aging effect and deteriorating maintenance. In order to counteract the aging effect on the machine, we assume that at most one maintenance is allowed throughout the planning horizon and the maintenance can be performed immediately after the processing of any job is completed. The maintenance duration is dependent on its starting time. The objective is to find jointly the optimal common slack time, the optimal maintenance position, and the optimal schedule such that the sum of the total earliness, the total tardiness, and the common slack time costs is minimized. We propose polynomial time algorithms for all the problems studied.

Abstract: In this paper higher order cone convex, pseudo convex, strongly pseudo convex, and quasiconvex functions are introduced. Higher order sufficient optimality conditions are given for a weak minimum, minimum, strong minimum and Benson proper minimum solution of a vector optimization problem. A higher order dual is associated and weak and strong duality results are established under these new generalized convexity assumptions.

Abstract: Sensitivity analysis and stability analysis in vector optimization are dealt with in this paper. First, some relationships between the second-order contingent derivative of a set-valued map and its profile map are obtained. Secondly, the upper semicontinuity and lower semicontinuity of second-order contingent derivatives of set-valued maps are established. Finally, by virtue of the second-order contingent derivative of set-valued maps, quantitative information and qualitative information on the behavior of the proper perturbation map are obtained.

Abstract: The paper is centered around a sum rule for the efficient (Pareto) \({\epsilon}\) -subdifferential of two convex vector mappings, having the property to be exact under a qualification condition. Such a formula has not been explored previously. Our formula which holds under the Attouch–Brezis as well as Moreau–Rockafellar conditions, reveals strangely a primordial presence of the convex (Fenchel) \({\epsilon}\) -subdifferential. This appearance turns out to be rather favorable. This effectively permits to derive approximate efficiency conditions in terms of Pareto subgradient and vectorial normal cone, which completely characterizes an \({\epsilon}\) -efficient solution in constrained convex vector optimization in (partially) ordered spaces. Our sum rule also allows a fundamental deduction of relation between Pareto and Fenchel \({\epsilon}\) -subdifferentials, which, in reality, brings out a certain gap linking \({\epsilon}\) -efficiency with \({\epsilon}\) -optimality. Scalarization approaches in connection with \({\epsilon}\) -subdifferentials are first established by simple proofs. This principle has contributed for a large part, not only for discovering the sum formula, but also for establishing some punctual necessary and/or sufficient conditions for Pareto \({\epsilon}\) -subdifferentiability.

Abstract: Revenue management can be used in many industries where there is a limited, perishable capacity and the market can be segmented. In this paper we focus on the sales of event tickets in the Sports and Entertainment industries, where tickets are sold exclusively as season tickets initially or as single events later in the selling horizon. We specifically study the optimal time to switch between these market segments dynamically as a function of the state of the system. Under Poisson demand processes, we find the optimal switching time is a set of time thresholds that depends on the remaining inventory and time left in the horizon. We use numerical experiments to show that significant profit improvements can be obtained by dynamically deciding the optimal switch time over the case when the date is announced in advance. We also study an extension where “early switch to a low-demand event” is allowed.

Abstract: This paper is concerned with a problem of maximizing the sum of several ratios of functions. We extend an algorithm, which has been designed to solve the sum-of-linear-ratios problem, for solving the sum-of-nonlinear-ratios problem. We also discuss the complexity of the problem and report the results of numerical experiments on the extended algorithm.

Abstract: In this paper, we develop a polynomial-time approximation scheme for the two-machine flow shop scheduling problem with several availability constraints on the first machine, under the resumable scenario.