The inverse optimal value problem

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost vectors, determine a cost vector such that the corresponding optimal objective value of the linear program is closest to the desired value. The above problem, referred here as the inverse optimal value problem, is significantly different from standard inverse optimization problems that involve determining a cost vector for a linear program such that a pre-specified solution vector is optimal. In this paper, we show that the inverse optimal value problem is NP-hard in general. We identify conditions under which the problem reduces to a concave maximization or a concave minimization problem. We provide sufficient conditions under which the associated concave minimization problem and, correspondingly, the inverse optimal value problem is polynomially solvable. For the case when the set of feasible cost vectors is polyhedral, we describe an algorithm for the inverse optimal value problem based on solving linear and bilinear programming problems. Some preliminary computational experience is reported.Mathematics Subject Classification (1999):49N45, 90C05, 90C25, 90C26, 90C31, 90C60Acknowledgement This research has been supported in part by the National Science Foundation under CAREER Award DMII-0133943. The authors thank two anonymous reviewers for valuable comments.     

Inverse Problems of Submodular Functions on Digraphs

In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficient vector c of the objective function, optimally and within bounds, such that x* becomes an optimal solution of the linear program. It is shown that the problem can be formulated as a combinatorial linear program and can be transformed further into a minimum cost circulation problem. Hence, it can be solved in strongly polynomial time. We also give a necessary and sufficient condition for the feasibility of the problem. Finally, we extend the discussion to the version of the inverse problem with multiple feasible solutions.     

On the adjustment problem for linear programs

We propose a generalization of the inverse problem which we will call the adjustment problem. For an optimization problem with linear objective function and its restriction defined by a given subset of feasible solutions, the adjustment problem consists in finding the least costly perturbations of the original objective function coefficients, which guarantee that an optimal solution of the perturbed problem is also feasible for the considered restriction. We describe a method of solving the adjustment problem for continuous linear programming problems when variables in the restriction are required to be binary.     

Constrained Inverse Minimum Spanning Tree Problems under the Bottleneck-Type Hamming Distance

In this paper, we consider the inverse minimum spanning tree problem under the bottleneck-type Hamming distance, where the weights of edges can be modified only within given intervals. We further consider the constrained case in which the total modification cost cannot exceed a given upper bound. It is shown that these inverse problems can be transformed into a minimum node cover problem on a bipartite graph, and we give a strongly polynomial time algorithm to solve this type of node cover problems. This work is supported by The National Natural Science Foundation of China (60021201), The Hong Kong Research Grant Council under the grant CERG 9040883 (CITYU 103003), and the Doctoral Foundation of Hohai University (2005-02).     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

On the dual of linear inverse problems

In linear inverse problems considered in this paper a vector with positive components is to be selected from a feasible set defined by linear constraints. The selection rule involves minimization of a certain function which is a measure of distance from a priori guess. Csiszar made an axiomatic approach towards defining a family of functions, we call it α-divergence, that can serve as logically consistent selection rules. In this paper we present an explicit and perfect dual of the resulting convex programming problem, prove the corresponding duality theorem and optimality criteria, and make some suggestions on an algorithmic solution.     

An exact solution framework for the minimum cost dominating tree problem

The minimum cost dominating tree problem is a recently introduced NP-hard problem, which consists of finding a tree of minimal cost in a given graph, such that for every node of the graph, the node or one of its neighbours is in the tree. We present an exact solution framework combining a primal–dual heuristic with a branch-and-cut approach based on a transformation of the problem into a Steiner arborescence problem with an additional constraint. The effectiveness of our approach is evaluated on testbeds proposed in literature containing instances with up to 500 nodes. Our framework manages to solve all but four instances from literature to proven optimality within 3 h (most of them in a few seconds). We provide optimal solution values for 69 instances from literature for which the optimal solution was previously unknown.     

The complete optimal stars-clustering-tree problem

We consider the following complete optimal stars-clustering-tree problem: Given a complete graph G=(V,E) with a weight on every edge and a collection of subsets of V, we want to find a minimum weight spanning tree T such that each subset of the vertices in the collection induces a complete star in T. One motivation for this problem is to construct a minimum cost (weight) communication tree network for a collection of (not necessarily disjoint) groups of customers such that each group induces a complete star. As a result the network will provide a “group broadcast” property, “group fault tolerance” and “group privacy”. We present another motivation from database systems with replications. For the case where the intersection graph of the subsets is connected we present a structure theorem that describes all feasible solutions. Based on it we provide a polynomial algorithm for finding an optimal solution. For the case where each subset induces a complete star minus at most k leaves we prove that the problem is NP-hard.     

Two level hierarchical time minimizing transportation problem

A two level hierarchical balanced time minimizing transportation problem is considered in this paper. The whole set of source-destination links consists of two disjoint partitions namely Level-I links and Level-II links. Some quantity of a homogeneous product is first shipped from sources to destinations by Level-I decision maker using only Level-I links, and on its completion the Level-II decision maker transports the remaining quantity of the product in an optimal fashion using only Level-II links. Transportation is assumed to be done in parallel in both the levels. The aim is to find that feasible solution for Level-I decision maker corresponding to which the optimal feasible solution for Level-II decision maker is such that the sum of shipment times in Level-I and Level-II is the least. To obtain the global optimal feasible solution of this non-convex optimization problem, related balanced time minimizing transportation problems are defined. Based upon the optimal feasible solutions of these related problems, standard cost minimizing transportation problems are constructed whose optimal feasible solutions provide various pairs for shipment times for Level-I and Level-II decision makers. The best out of these pairs is finally selected. Being dependent upon solutions of a finite number of balanced time minimizing and cost minimizing transportation problems, the proposed algorithm is a polynomial bound algorithm. The developed algorithm has been implemented and tested on a variety of test problems and performance is found to be quite encouraging.     

On accuracy,robustness and tolerances in vector Boolean optimization

A Boolean programming problem with a finite number of alternatives where initial coefficients (costs) of linear payoff functions are subject to perturbations is considered. We define robust solution as a feasible solution which for a given set of realizations of uncertain parameters guarantees the minimum value of the worst-case relative regret among all feasible solutions. For the Pareto optimality principle, an appropriate definition of the worst-case relative regret is specified. It is shown that this definition is closely related to the concept of accuracy function being recently intensively studied in the literature. We also present the concept of robustness tolerances of a single cost vector. The tolerance is defined as the maximum level of perturbation of the cost vector which does not destroy the solution robustness. We present formulae allowing the calculation of the robustness tolerance obtained for some initial costs. The results are illustrated with several numerical examples.     

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

We consider the separable nonlinear and strictly convex single-commodity network flow problem (SSCNFP). We develop a computational scheme for generating a primal feasible solution from any Lagrangian dual vector; this is referred to as “early primal recovery”. It is motivated by the desire to obtain a primal feasible vector before convergence of a Lagrangian scheme; such a vector is not available from a Lagrangian dual vector unless it is optimal. The scheme is constructed such that if we apply it from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal (feasible) vectors converge to the unique optimal primal flow vector. It is therefore also a convergent Lagrangian heuristic, akin to those primarily devised within the field of combinatorial optimization but with the contrasting and striking advantage that it is guaranteed to yield a primal optimal solution in the limit. Thereby we also gain access to a new stopping criterion for any Lagrangian dual algorithm for the problem, which is of interest in particular if the SSCNFP arises as a subproblem in a more complex model. We construct instances of convergent Lagrangian heuristics that are based on graph searches within the residual graph, and therefore are efficiently implementable; in particular we consider two shortest path based heuristics that are based on the optimality conditions of the original problem. Numerical experiments report on the relative efficiency and accuracy of the various schemes.     

Capacitated,balanced, sequential location-allocation problems on chains and trees

This paper considers finite horizon, multiperiod, sequential, minisum location-allocation problems on chain graphs and tree networks. The demand has both deterministic and probabilistic components, and increases dynamically from period to period. The problem is to locate one additionalcapacitated facility in each of thep specified periods, and to determine the service allocations of the facilities, in order to optimally satisfy the demand on the network. In this context, two types of objective criteria or location strategies are addressed. The first is a myopic strategy in which the present period cost is minimized sequentially for each period, and the second is a discounted present worth strategy. For the chain graph, we analyze ap-facility problem under both these criteria, while for the tree graph, we analyze a 3-facility myopic problem, and a 2-facility discounted present worth problem. All these problems are nonconvex, and we specify a finite set of candidate solutions which may be compared in order to determine a global optimal solution.     

The geometric generalized minimum spanning tree problem with grid clustering

This paper is concerned with a special case of the generalized minimum spanning tree problem. The problem is defined on an undirected graph, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges. The problem is to find a tree of minimum cost containing at least one vertex in each cluster. We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance defines the edge cost. We prove that the problem is strongly -hard even in the case of a special structure of the clustering called grid clustering. We construct an exact exponential time dynamic programming algorithm and, based on this dynamic programming algorithm, we develop a polynomial time approximation scheme for the problem with grid clustering.     

Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm

We present an exact algorithm for solving the generalized minimum spanning tree problem (GMST). Given an undirected connected graph and a partition of the graph vertices, this problem requires finding a least-cost subgraph spanning at least one vertex out of every subset. In this paper, the GMST is formulated as a minimum spanning tree problem with side constraints and solved exactly by a branch-and-bound algorithm. Lower bounds are derived by relaxing, in a Lagrangian fashion, complicating constraints to yield a modified minimum cost spanning tree problem. An efficient preprocessing algorithm is implemented to reduce the size of the problem. Computational tests on a large set of randomly generated instances with as many as 250 vertices, 1000 edges, and 25 subsets provide evidence that the proposed solution approach is very effective.     

A Polynomial Time Approximation Scheme for Optimal Product-Requirement Communication Spanning Trees

Given an undirected graph with nonnegative edge lengths and nonnegative vertex weights, the routing requirement of a pair of vertices is assumed to be the product of their weights. The routing cost for a pair of vertices on a given spanning tree is defined as the length of the path between them multiplied by their routing requirement. The optimal product-requirement communication spanning tree is the spanning tree with minimum total routing cost summed over all pairs of vertices. This problem arises in network design and computational biology. For the special case that all vertex weights are identical, it has been shown that the problem is NP-hard and that there is a polynomial time approximation scheme for it. In this paper we show that the generalized problem also admits a polynomial time approximation scheme.     

Lower and upper bounds for the spanning tree with minimum branch vertices

We study a variant of the spanning tree problem where we require that, for a given connected graph, the spanning tree to be found has the minimum number of branch vertices (that is vertices of the tree whose degree is greater than two). We provide four different formulations of the problem and compare different relaxations of them, namely Lagrangian relaxation, continuous relaxation, mixed integer-continuous relaxation. We approach the solution of the Lagrangian dual both by means of a standard subgradient method and an ad-hoc finite ascent algorithm based on updating one multiplier at the time. We provide numerical result comparison of all the considered relaxations on a wide set of benchmark instances. A useful follow-up of tackling the Lagrangian dual is the possibility of getting a feasible solution for the original problem with no extra costs. We evaluate the quality of the resulting upper bound by comparison either with the optimal solution, whenever available, or with the feasible solution provided by some existing heuristic algorithms.     

Inverse multi-objective combinatorial optimization

Inverse multi-objective combinatorial optimization consists of finding a minimal adjustment of the objective functions coefficients such that a given set of feasible solutions becomes efficient. An algorithm is proposed for rendering a given feasible solution into an efficient one. This is a simplified version of the inverse problem when the cardinality of the set is equal to one. The adjustment is measured by the Chebyshev distance. It is shown how to build an optimal adjustment in linear time based on this distance, and why it is right to perform a binary search for determining the optimal distance. These results led us to develop an approach based on the resolution of mixed-integer linear programs. A second approach based on a branch-and-bound is proposed to handle any distance function that can be linearized. Finally, the initial inverse problem is solved by a cutting plane algorithm.     

Some equivalent problems in set optimization

Given a set-valued optimization problem (P), there is more than one way of defining the solutions associated with it. Depending on the decision maker’s preference, we consider the vector criterion or the set criterion. Both criteria of solution are considered together to solve problem (P) by reducing the feasible set.     

Determining distributed parameters in a neuronal cable model on a tree graph

For graph domains without cycles, we show how unknown coefficients and source terms for a parabolic equation can be recovered from the dynamical Neumann‐to‐Dirichlet map associated with the boundary vertices. Through use of a companion wave equation problem, the topology of the tree graph, degree of the vertices, and edge lengths can also be recovered. The motivation for this work comes from a neuronal cable equation defined on the neuron's dendritic tree, and the inverse problem concerns parameter identification of k unknown distributed conductance parameters. Copyright © 2017 John Wiley & Sons, Ltd.     

Uniform-cost inverse absolute and vertex center location problems with edge length variations on trees

This article considers the inverse absolute and the inverse vertex 1-center location problems with uniform cost coefficients on a tree network T with n+1 vertices. The aim is to change (increase or reduce) the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified vertex s becomes an absolute (or a vertex) 1-center under the new edge lengths. First an O(nlogn) time method for solving the height balancing problem with uniform costs is described. In this problem the height of two given rooted trees is equalized by decreasing the height of one tree and increasing the height of the second rooted tree at minimum cost. Using this result a combinatorial O(nlogn) time algorithm is designed for the uniform-cost inverse absolute 1-center location problem on tree T. Finally, the uniform-cost inverse vertex 1-center location problem on T is investigated. It is shown that the problem can be solved in O(nlogn) time if all modified edge lengths remain positive. Dropping this condition, the general model can be solved in O(r_vnlogn) time where the parameter r_v is bounded by ⌈n/2⌉. This corrects an earlier result of Yang and Zhang.