Abstract: Abstract We introduce and study the notion of the \(\left( \sigma ,y\right) \) -conjugate of a proper \(\sigma \) -convex function. Some relations between the \(\sigma \) -subdifferentials and the Clarke–Rockafellar subdifferential are established. Also, we present some results regarding the \(\sigma \) -monotonicity of the \(\sigma \) -subdifferential of a function and its \(\sigma \) -convexity. Moreover, we obtain some particular relationships between the \(\sigma \) -subdifferential and the \(\left( \sigma ,y\right) \) -conjugate. PubDate: 2021-01-16

Abstract: Abstract Recently, the estimation problem of upper and lower bounds of the solution set of the tensor complementarity problem has been studied when the tensor involved is a strictly semi-positive tensor or one of its subclasses. This paper aims to study such an estimation problem in a larger scope. First, we propose a lower bound formula under the condition that the tensor complementarity problem has a solution. When the problem under consideration falls back to several types of problems that have been studied, the achieved result improves the relevant known results. Second, by means of a newly introduced quantity, we give an upper bound formula of the solution set when the problem has a solution and the tensor involved is an \(R_0\) -tensor. This formula is new even when the concerned problem falls back to several problems that have already been discussed. Several examples are also given to confirm our theoretical findings. PubDate: 2021-01-12

Abstract: Abstract This work proposes a bi-objective mathematical optimization model and a two-stage heuristic for a real-world application of the heterogeneous Dynamic Dial-a-Ride Problem with no rejects, i.e., a patient transportation system. The problem consists of calculating route plans to meet a set of transportation requests by using a given heterogeneous vehicle fleet. These transportation requests can be either static or dynamic, and all of them must be attended to. In the first stage of the proposed heuristic, the problem’s static part is solved by applying a General Variable neighborhood Search based algorithm. In the second stage, the dynamic requests are dealt with by implementing a simple insertion heuristic. We create different instances based on the real data provided by a Brazilian city’s public health care system and test the proposed approach on them. The analysis of the results shows that the higher the level of dynamism, i.e., the number of urgent requests on each instance, the smaller the objective function value will be in the static part. The results also demonstrate that a higher level of dynamism increases the chance of a time window violation happening. Besides, we use the weighted sum method of the two conflicting objectives to analyze the trade-off between them and create an approximation for the Pareto frontier. PubDate: 2021-01-08

Abstract: Abstract Beside of cryptography-the primary traditional methods for ensuring information security and confidentiality, the appearance of the physical layer security approach plays an important role for not only enabling the data transmission confidentially without relying on higher-layer encryption, but also enhancing confidentiality of the secret key distribution in cryptography. Many techniques are employed in physical layers to improve secure transmission including cooperative relaying and beamforming technique. In this paper, we consider the secrecy rate maximization problems using two techniques mentioned above with two different relaying protocols: Amplify-and-Forward and Decode-and-Forward. The optimization problems with the aim of maximizing secrecy rate subject to total and individual relay power constraints are formulated as nonconvex problems, which can be reformulated as DC (difference of two convex functions) programs and thus can be solved by DC Algorithms (DCA). The special structure of feasible set is exploited which results to an efficient DC decomposition in the sense that it leads to convex subproblems that can be explicitly solved. The numerical results show that the proposed DCA schemes are better than the existing methods in terms of both runtime and secrecy rate. PubDate: 2021-01-08

Abstract: Abstract Sentence compression is an important problem in natural language processing with wide applications in text summarization, search engine and human–AI interaction system etc. In this paper, we design a hybrid extractive sentence compression model combining a probability language model and a parse tree language model for compressing sentences by guaranteeing the syntax correctness of the compression results. Our compression model is formulated as an integer linear programming problem, which can be rewritten as a difference-of-convex (DC) programming problem based on the exact penalty technique. We use a well-known efficient DC algorithm—DCA to handle the penalized problem for local optimal solutions. Then a hybrid global optimization algorithm combining DCA with a parallel branch-and-bound framework, namely PDCABB, is used for finding global optimal solutions. Numerical results demonstrate that our sentence compression model can provide excellent compression results evaluated by F-score, and indicate that PDCABB is a promising algorithm for solving our sentence compression model. PubDate: 2021-01-07

Abstract: Abstract In this paper, we study the extended trust-region subproblem in which the trust-region intersects the ball with m linear inequality constraints (m-eTRS). We assume that the linear constraints do not intersect inside the ball. We show that the optimal solution of m-eTRS can be found by solving one TRS, computing the local non-global minimizer of TRS if it exists and solving at most two TRSs with an additional linear equality constraint (1-eqTRS). Both TRS and (1-eqTRS) are polynomially and efficiently solvable, thus the new algorithm significantly improves over the SOCP/SDP relaxation of Burer and Yang [Math Program 149(1-2):253–264, 2015]. on two classes of test problems, the efficiency of the proposed approach is compared with the SOCP/SDP relaxation and branch and bound algorithm of Beck and Pan [J Global Optim 69(2):309–342, 2017]. PubDate: 2021-01-06

Abstract: Abstract In this paper, we examine the properly twice epi-differentiability and compute the second order epi-subderivative of the indicator function to a class of sets including the finite union of parabolically derivable and parabolically regular sets. In this way, we provide no-gap second order optimality conditions for a disjunctive constrained problem. Moreover, we derive applications of our results to some types of disjunctive programs. PubDate: 2021-01-06

Abstract: Abstract The vertex colourability problem is to determine, for a given graph and a given natural k, whether it is possible to split the graph’s vertex set into at most k subsets, each of pairwise non-adjacent vertices, or not. A hereditary class is a set of simple graphs, closed under deletion of vertices. Any such a class can be defined by the set of its forbidden induced subgraphs. For all but four hereditary classes, defined by a pair of connected five-vertex induced prohibitions, the complexity status of the vertex colourability problem is known. In this paper, we reduce the number of the open cases to three, by showing polynomial-time solvability of the problem for \(\{claw,butterfly\}\) -free graphs. A claw is the star graph with three leaves, a butterfly is obtained by coinciding a vertex in a triangle with a vertex in another triangle. PubDate: 2021-01-06

Abstract: Abstract We consider the maximum shortest path interdiction problem by upgrading edges on trees under Hamming distance (denoted by (MSPITH)), which has wide applications in transportation network, network war and terrorist network. The problem (MSPITH) aims to maximize the length of the shortest path from the root of a tree to all its leaves by upgrading edge weights such that the upgrade cost under sum-Hamming distance is upper-bounded by a given value. We show that the problem (MSPITH) under weighted sum-Hamming distance is NP-hard. We consider two cases of the problem (MSPITH) under unit sum-Hamming distance based on the number K of critical edges. We propose a greedy algorithm within \(O(n+l\log l)\) time when \(K=1\) and a dynamic programming algorithm within \(O(n(\log n+K^3))\) time when \(K>1\) , where n and l are the numbers of nodes and leaves in a tree, respectively. Furthermore, we consider a minimum cost shortest path interdiction problem by upgrading edges on trees under unit Hamming distance, denoted by (MCSPITUH) and propose a binary search algorithm within \(O(n^4\log n)\) time, where a dynamic programming algorithm is executed in each iteration to solve its corresponding problem (MSPITH). Finally, we design numerical experiments to show the effectiveness of the algorithms. PubDate: 2021-01-06

Abstract: Abstract In this paper, we study the absolute value equation (AVE) \(Ax-b= x \) . One effective approach to handle AVE is by using concave minimization methods. We propose a new method based on concave minimization methods. We establish its finite convergence under mild conditions. We also study some classes of AVEs which are polynomial time solvable. PubDate: 2021-01-05

Abstract: Abstract The paper deals with a Berge equilibrium (Théorie générale des jeux à-personnes, Gauthier Villars, Paris, 1957; Some problems of non-antagonistic differential games, 1985) in the bimatrix game for mixed strategies. Motivated by Nash equilibrium (Ann Math 54(2):286, 1951; Econometrica 21(1):128–140, 1953), we prove an existence of Berge equilibrium in the bimatrix game. Based on Mills theorem (J Soc Ind Appl Math 8(2):397–402, 1960), we reduce the bimatrix game to a nonconvex optimization problem. We illustrate the proposed approach on an example. PubDate: 2021-01-04

Abstract: Abstract During major infectious disease outbreak, such as COVID-19, the goods and parcels supply and distribution for the isolated personnel has become a key issue worthy of attention. In this study, we propose a delivery problem that arises in the last-mile delivery during major infectious disease outbreak. The problem is to construct a Hamiltonian tour over a subset of candidate parking nodes, and each customer is assigned to the nearest parking node on the tour to pick up goods or parcels. The aim is to minimize the total cost, including the routing, allocation, and parking costs. We propose three models to formulate the problem, which are node-based, flow-based and bilevel programing formulations. Moreover, we develop a variable neighborhood search algorithm based on the ideas from the bilevel programing formulations to solve the problem. Finally, the proposed algorithm is tested on a set of randomly generated instances, and the results indicate the effectiveness of the proposed approach. PubDate: 2021-01-04

Abstract: Abstract In the sequel, we outline necessary and sufficient condition to the existence of extremas of a function on a self-similar set, and we describe discrete gradient algorithm to find the extrema. PubDate: 2021-01-04

Abstract: Abstract The Fixed-Tree BMEP (FT-BMEP) is a special case of the Balanced Minimum Evolution Problem (BMEP) that consists of finding the assignment of a set of n taxa to the n leaves of a given unrooted binary tree so as to minimize the BMEP objective function. Deciding the computational complexity of the FT-BMEP has been an open problem for almost a decade. Here, we show that a few modifications to Fiorini and Joret’s proof of the \(\mathcal {NP}\) -hardness of the BMEP suffice to prove the general \(\mathcal {NP}\) -hardness of the FT-BMEP as well as its strong inapproximability. PubDate: 2021-01-02

Abstract: Abstract The purpose of this paper is to introduce a new modified subgradient extragradient method for finding an element in the set of solutions of the variational inequality problem for a pseudomonotone and Lipschitz continuous mapping in real Hilbert spaces. It is well known that for the existing subgradient extragradient methods, the step size requires the line-search process or the knowledge of the Lipschitz constant of the mapping, which restrict the applications of the method. To overcome this barrier, in this work we present a modified subgradient extragradient method with adaptive stepsizes and do not require extra projection or value of the mapping. The advantages of the proposed method only use one projection to compute and the strong convergence proved without the prior knowledge of the Lipschitz constant of the inequality variational mapping. Numerical experiments illustrate the performances of our new algorithm and provide a comparison with related algorithms. PubDate: 2021-01-02

Abstract: Abstract In this paper, we present the problem in which a municipal company operating in the waste management sector willing to encourage users to use differentiated waste collection facilities, designs a utility user function to attain such a goal. The problem is modeled in terms of bilevel optimization where the leader is the municipal firm which aims at maximizing the concurrent fraction of user waste demand thrown in recycling facilities, and the follower are users aiming at maximizing the utility function proposed by the leader. The resulting bilevel model is analysed in terms of stability showing that its optimistic solution value equals the pessimistic solution value. A solution approach is presented. Finally, a computational study shows the effectiveness of our proposal. PubDate: 2021-01-02

Abstract: Abstract Researchers and practitioners have addressed many variants of facility locations problems. Each location problem can be substantially different from each other depending on the objectives and/or constraints considered. In this paper, the bi-objective obnoxious p-median problem (Bi-OpM) is addressed given the huge interest to locate facilities such as waste or hazardous disposal facilities, nuclear power or chemical plants and noisy or polluting services, among others. The Bi-OpM aims to locate p facilities maximizing two different objectives: the distance between each customer and their nearest facility center and the dispersion among facilities. To address the Bi-OpM problem a Multi-objective Parallel Variable Neighborhood Search approach (Mo-PVNS) is implemented. Computational results indicate the superiority of the Mo-PVNS compared to the state-of-art algorithms. PubDate: 2021-01-02

Abstract: Abstract In this paper, we present necessary/sufficient conditions for the convex polynomial programming (CPP) problems. Some new stability results for parametric CPP problems are characterized under a regular condition. We give a positive answer for the open question in Kim et al. (Optim Lett 6:363–373, 2012) for the solution existence of convex quadratic programming problems. PubDate: 2021-01-02

Abstract: Abstract The purpose of this short note is to show that the Christoffel–Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis. Geometric interpretations and algorithmic consequences are mentioned. PubDate: 2021-01-01

Abstract: Abstract This paper examines a new model for hub location known as the hub location routing problem. The problem shares similarities with the well studied uncapacitated single allocation p-hub median problem except that the hubs are now connected to each other by a cyclical path (or tour) known as the global route, each cluster of non-hub nodes and assigned hub is also connected by a single tour known as a local route, and the length of the local routes is constrained to a maximum number of nodes. Thus, aside from the normal tasks of hub selection and allocation of non-hub nodes, up to p + 1 travelling salesman problems need to be solved. A heuristic based on the general variable neighborhood search framework is proposed here to solve this very complicated problem. The improvement phase of the algorithm uses a sequential variable neighborhood descent with multiple neighborhoods required to suit the complex nature of the problem. A best sequencing of the neighborhoods is established through empirical testing. The perturbation phase known as the shaking procedure also uses a well-structured selection of neighborhoods in order to effectively diversify the search to different regions of the solution space. Extensive computational testing shows that the new heuristic significantly outperforms the state-of-the-art. Out of 912 test instances from the literature, we are able to obtain 691 new best known solutions. Not only are the improvements in objective values quite impressive, but also these new solutions are obtained in a small fraction of the time required by the competing algorithms. PubDate: 2020-11-28