Cauchy problem for a class of integro-differential equations

We consider the Cauchy problem for integro-differential equations containing Wiener-Hopf operators. We define the characteristic polynomial of the problem. We show that if all roots of the characteristic polynomial have negative imaginary parts and the corresponding strong moment problem is solvable, then the problem is equivalent to a Wiener-Hopf integral equation. We consider an example of an application of the result to a problem related to nonlocal wave interaction.     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

Preemptive repayment policy for multiple loans

We study the multi-loan-repayment problem by formulating it as a single-machine scheduling problem with preemptive and time-dependent processing times to minimize the makespan. We transform the scheduling problem into a continuous non-linear optimization problem and obtain an approximate solution by solving a series of the corresponding linear programming problems. We also identify agreeable conditions for the problem and discuss the computational complexity of the problem.     

An Interactive Algorithm for Multi-objective Route Planning

We address the route selection problem for Unmanned Air Vehicles (UAV) under multiple objectives. We consider a general case for this problem, where the UAV has to visit several targets and return to the base. We model this problem as a combination of two combinatorial problems. First, the path to be followed between each pair of targets should be determined. We model this as a multi-objective shortest path problem. Additionally, we need to determine the order of the targets to be visited. We model this as a multi-objective traveling salesperson problem (MOTSP). The overall problem is a combination of these two problems, which we define as a generalized MOTSP. We develop an exact interactive approach to identify the best paths and the best tour of a decision maker under a linear utility function.     

Product sequencing and pricing under cascade browse model

We study the assortment optimization problem under the newly proposed cascade browse model. We propose a constant approximate solution to this problem. As a byproduct, we propose the first fully polynomial-time approximation scheme (FPTAS) for the classic assortment optimization problem subject to one capacity constraint and one cardinality constraint. We also studied a joint pricing and sequencing problem under the above model and develop a constant approximate solution to this problem.     

Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

A dynamic uncapacitated lot-sizing problem with co-production

We consider an extension of the dynamic uncapacitated lot-sizing problem to account for co-production, where multiple products are produced simultaneously in a single production run. We formulate the problem as a mixed-integer linear programming problem. We then show that a variant of the well-known zero-inventory property holds for this problem, and use this property to extend a dynamic program given for the single-item lot-sizing to solve the problem with co-production. Finally, we provide an illustrative example for our approach.     

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

On the inverse problem of linear programming and its application to minimum weight perfect k-matching

We first introduce the inverse problem of linear programming. We then apply it to the inverse problem of minimum weight perfect k-matching of bipartite graphs. We show that there is a strongly polynomial algorithm for solving the problem.     

Covering partially directed graphs with directed paths

We consider graphs which contain both directed and undirected edges (partially directed graphs). We show that the problem of covering the edges of such graphs with a minimum number of edge-disjoint directed paths respecting the orientations of the directed edges is polynomially solvable. We exhibit a good characterization for this problem in the form of a min-max theorem. We introduce a more general problem including weights on possible orientations of the undirected edges. We show that this more general weighted formulation is equivalent to the weighted bipartite b-factor problem. This implies the existence of a strongly polynomial algorithm for this weighted generalization of Euler's problem to partially directed graphs (compare this with the negative results for the mixed Chinese postman problem). We also provide a compact linear programming formulation for the weighted generalization that we propose.     

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

Levitin–Polyak well-posedness by perturbations of split minimization problems

In this paper, we extend well-posedness notions to the split minimization problem which entails finding a solution of one minimization problem such that its image under a given bounded linear transformation is a solution of another minimization problem. We prove that the split minimization problem in the setting of finite-dimensional spaces is Levitin–Polyak well-posed by perturbations provided that its solution set is nonempty and bounded. We also extend well-posedness notions to the split inclusion problem. We show that the well-posedness of the split convex minimization problem is equivalent to the well-posedness of the equivalent split inclusion problem.     

A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.     

On the robust assignment problem under a fixed number of cost scenarios

We investigate the complexity of the min-max assignment problem under a fixed number of scenarios. We prove that this problem is polynomial-time equivalent to the exact perfect matching problem in bipartite graphs, an infamous combinatorial optimization problem of unknown computational complexity.     

A packet filter placement problem with application to defense against spoofed denial of service attacks

We analyze a problem in computer network security, wherein packet filters are deployed to defend a network against spoofed denial of service attacks. Information on the Internet is transmitted by the exchange of IP packets, which must declare their origin and destination addresses. A route-based packet filter verifies whether the purported origin of a packet is correct with respect to the current route map. We examine the optimization problem of finding a minimum cardinality set of nodes to filter in the network such that no spoofed packet can reach its destination. We prove that this problem is NP-hard, and derive properties that explicitly relate the filter placement problem to the vertex cover problem. We identify topologies and routing policies for which a polynomial-time solution to the minimum filter placement problem exists, and prove that under certain routing conditions a greedy heuristic for the filter placement problem yields an optimal solution.     

The complexity of recursive constraint satisfaction problems

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem.     

Problem on small motions of ideal rotating relaxing fluid

We study an evolution problem on small motions of the ideal rotating relaxing fluid in bounded domains. We begin from the problem posing. Then we reduce the problem to a second-order integrodifferential equation in a Hilbert space. Using this equation, we prove a strong unique solvability problem for the corresponding initial-boundary value problem.     

Modified Moment Problem in Complex Domains

We study the moment problem in a complex domain. The results obtained generalize and sharpen the well-known Kaz'min result. We also consider applications of these results to the Abel--Goncharov problem (the uniqueness and construction problem) and to the completeness problem for systems of analytic functions.     

Mixed equilibrium problems with Z*-bifunctions and least element problems in Banach lattices

The purpose of this paper is to study the relations among a mixed equilibrium problem, a least element problem and a minimization problem in Banach lattices. We propose the concept of Z*-bifunctions as well as the concept of a feasible set for the mixed equilibrium problem. We prove that the feasible set of the mixed equilibrium problem is a sublattice provided that the associated bifunction is a strictly α-monotone Z*-bifunction. We establish the equivalence of the mixed equilibrium problem, the least element problem and the minimization problem under strict α-monotonicity and Z*-bifunction conditions.     

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.