On an inverse linear programming problem

A method for solving the following inverse linear programming (LP) problem is proposed. For a given LP problem and one of its feasible vectors, it is required to adjust the objective function vector as little as possible so that the given vector becomes optimal. The closeness of vectors is estimated by means of the Euclidean vector norm. The inverse LP problem is reduced to a problem of unconstrained minimization for a convex piecewise quadratic function. This minimization problem is solved by means of the generalized Newton method.     

Some inverse optimization problems under the Hamming distance

Given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph, the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal. The aim is to optimize the difference between the initial cost vector and the disturbed one. This difference can be measured in several ways. We consider the Hamming distance measuring in how many components two vectors are different, where weights are associated to the components. General algorithms for the bottleneck or minimax criterion are described and (after modification) applied to the inverse minimum spanning tree problem, the inverse shortest path tree problem and the linear assignment problem.     

Inverse optimization for linearly constrained convex separable programming problems

In this paper, we study inverse optimization for linearly constrained convex separable programming problems that have wide applications in industrial and managerial areas. For a given feasible point of a convex separable program, the inverse optimization is to determine whether the feasible point can be made optimal by adjusting the parameter values in the problem, and when the answer is positive, find the parameter values that have the smallest adjustments. A sufficient and necessary condition is given for a feasible point to be able to become optimal by adjusting parameter values. Inverse optimization formulations are presented with ℓ₁ and ℓ₂ norms. These inverse optimization problems are either linear programming when ℓ₁ norm is used in the formulation, or convex quadratic separable programming when ℓ₂ norm is used.     

On robust optimization of two-stage systems

Robust-optimization models belong to a special class of stochastic programs, where the traditional expected cost minimization objective is replaced by one that explicitly addresses cost variability. This paper explores robust optimization in the context of two-stage planning systems. We show that, under arbitrary measures for variability, the robust optimization approach might lead to suboptimal solutions to the second-stage planning problem. As a result, the variability of the second-stage costs may be underestimated, thereby defeating the intended purpose of the model. We propose sufficient conditions on the variability measure to remedy this problem. Under the proposed conditions, a robust optimization model can be efficiently solved using a variant of the L-shaped decomposition algorithm for traditional stochastic linear programs. We apply the proposed framework to standard stochastic-programming test problems and to an application that arises in auctioning excess electric power. Mathematics Subject Classification (1991):90C15, 90C33, 90B50, 90A09, 90A43Supported in part by NSF Grants DMI-0099726 and DMI-0133943     

求解一类逆鲁棒优化问题的牛顿型扰动方法

逆优化问题是指通过调整目标函数和约束中的某些参数使得已知的一个解成为参数调整后的优化问题的最优解.本文考虑求解一类逆鲁棒优化问题.首先,我们将该问题转化为带有一个线性等式约束,一个二阶锥互补约束和一个线性互补约束的极小化问题;其次,通过一类扰动方法来对转化后的极小化问题进行求解,然后利用带Armijo线搜索的非精确牛顿法求解每一个扰动问题.最后,通过数值实验验证该方法行之有效.     

A further study on inverse linear programming problems

In this paper we continue our previous study (Zhang and Liu, J. Comput. Appl. Math. 72 (1996) 261–273) on inverse linear programming problems which requires us to adjust the cost coefficients of a given LP problem as less as possible so that a known feasible solution becomes the optimal one. In particular, we consider the cases in which the given feasible solution and one optimal solution of the LP problem are 0–1 vectors which often occur in network programming and combinatorial optimization, and give very simple methods for solving this type of inverse LP problems. Besides, instead of the commonly used l₁ measure, we also consider the inverse LP problems under l_∞ measure and propose solution methods.     

Linear programming with spheres and hemispheres of objective vectors

Given a linear program with a boundedp-dimensional feasible region let the objective vector range over ans-sphere, that is, ans-dimensional sphere centered at the origin wheres does not exceedp–1. If the feasible region and the sphere are in general position with respect to each other, then the corresponding set of all optimal solutions is a topologicals-sphere. Similar results are developed for unbounded feasible regions and hemispheres of objective vectors.This research is based on work supported in part by the National Science Foundation under Grant DMS-86-03232.     

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.     

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.     

Minimal concave cost rebalance of a portfolio to the efficient frontier

One usually constructs a portfolio on the efficient frontier, but it may not be efficient after, say three months since the efficient frontier will shift as the elapse of time. We then have to rebalance the portfolio if the deviation is no longer acceptable. The method to be proposed in this paper is to find a portfolio on the new efficient frontier such that the total transaction cost required for this rebalancing is minimal. This problem results in a nonconvex minimization problem, if we use mean-variance model. In this paper we will formulate this problem by using absolute deviation as the measure of risk and solve the resulting linearly constrained concave minimization problem by a branch and bound algorithm successfully applied to portfolio optimization problem under concave transaction costs. It will be demonstrated that this method is efficient and that it leads to a significant reduction of transaction costs. Key words.portfolio optimization – rebalance – mean-absolute deviation model – concave cost minimization – optimization over the efficient set – global optimizationMathematics Subject Classification (1991):20E28, 20G40, 20C20     

A Bilinear Algorithm for Optimizing a Linear Function over the Efficient Set of a Multiple Objective Linear Programming Problem

The problem Q of optimizing a linear function over the efficient set of a multiple objective linear program serves several useful purposes in multiple criteria decision making. However, Q is in itself a difficult global optimization problem, whose local optima, frequently large in number, need not be globally optimal. Indeed, this is due to the fact that the feasible region of Q is, in general, a nonconvex set. In this paper we present a monotonically increasing algorithm that finds an exact, globally-optimal solution for Q. Our approach does not require any hypothesis on the boundedness of neither the efficient set EP nor the optimal objective value. The proposed algorithm relies on a simplified disjoint bilinear program that can be solved through the use of well-known specifically designed methods within nonconvex optimization. The algorithm has been implemented in C and preliminary numerical results are reported.     

Lagrangian decomposition of block-separable mixed-integer all-quadratic programs

The purpose of this paper is threefold. First we propose splitting schemes for reformulating non-separable problems as block-separable problems. Second we show that the Lagrangian dual of a block-separable mixed-integer all-quadratic program (MIQQP) can be formulated as an eigenvalue optimization problem keeping the block-separable structure. Finally we report numerical results on solving the eigenvalue optimization problem by a proximal bundle algorithm applying Lagrangian decomposition. The results indicate that appropriate block-separable reformulations of MIQQPs could accelerate the running time of dual solution algorithms considerably.The work was supported by the German Research Foundation (DFG) under grant NO 421/2-1Mathematics Subject Classification (2000): 90C22, 90C20, 90C27, 90C26, 90C59     

Descent methods for convex essentially smooth minimization

Consider the problem of minimizing a convex essentially smooth function over a polyhedral set. For the special case where the cost function is strictly convex, we propose a feasible descent method for this problem that chooses the descent directions from a finite set of vectors. When the polyhedral set is the nonnegative orthant or the entire space, this method reduces to a coordinate descent method which, when applied to certain dual of linearly constrained convex programs with strictly convex essentially smooth costs, contains as special cases a number of well-known dual methods for quadratic and entropy (either –logx orx logx) optimization. Moreover, convergence of these dual methods can be inferred from a general convergence result for the feasible descent method. When the cost function is not strictly convex, we propose an extension of the feasible descent method which makes descent along the elementary vectors of a certain subspace associated with the polyhedral set. The elementary vectors are not stored, but generated using the dual rectification algorithm of Rockafellar. By introducing an -complementary slackness mechanism, we show that this extended method terminates finitely with a solution whose cost is within an order of of the optimal cost. Because it uses the dual rectification algorithm, this method can exploit the combinatorial structure of the polyhedral set and is well suited for problems with a special (e.g., network) structure.This work was partially supported by the US Army Research Office Contract No. DAAL03-86-K-0171 and by the National Science Foundation Grant No. ECS-85-19058.     

Max-min sum minimization transportation problem

A non-convex optimization problem involving minimization of the sum of max and min concave functions over a transportation polytope is studied in this paper. Based upon solving at most (g+1)(< p) cost minimizing transportation problems with m sources and n destinations, a polynomial time algorithm is proposed which minimizes the concave objective function where, p is the number of pairwise disjoint entries in the m× n time matrix {t _ij } sorted decreasingly and T ^g is the minimum value of the max concave function. An exact global minimizer is obtained in a finite number of iterations. A numerical illustration and computational experience on the proposed algorithm is also included. We are thankful to Prof. S. N. Kabadi, University of New Brunswick-Fredericton, Canada, who initiated us to the type of problem discussed in this paper. We are also thankful to Mr. Ankit Khandelwal, Ms. Neha Gupta and Ms. Anuradha Beniwal, who greatly helped us in the implementation of the proposed algorithm.     

范数下无容量限制设施选址逆问题的求解方法

一个优化问题的逆问题是这样一类问题,在给定该优化问题的一个可行解时,通过最小化目标函数中参数的改变量(在某个范数下)使得该可行解成为改变参数后的该优化问题的最优解。对于本是NP-难问题的无容量限制设施选址问题,证明了其逆问题仍是NP-难的。研究了使用经典的行生成算法对无容量限制设施选址的逆问题进行计算,并给出了求得逆问题上下界的启发式方法。两种方法分别基于对子问题的线性松弛求解给出上界和利用邻域搜索以及设置迭代循环次数的方式给出下界。数值结果表明线性松弛法得到的上界与最优值差距较小,但求解效率提升不大;而启发式方法得到的下界与最优值差距极小,极大地提高了求解该逆问题的效率。     

Generalized Benders decomposition

J. F. Benders devised a clever approach for exploiting the structure of mathematical programming problems withcomplicating variables (variables which, when temporarily fixed, render the remaining optimization problem considerably more tractable). For the class of problems specifically considered by Benders, fixing the values of the complicating variables reduces the given problem to an ordinary linear program, parameterized, of course, by the value of the complicating variables vector. The algorithm he proposed for finding the optimal value of this vector employs a cutting-plane approach for building up adequate representations of (i) the extremal value of the linear program as a function of the parameterizing vector and (ii) the set of values of the parameterizing vector for which the linear program is feasible. Linear programming duality theory was employed to derive the natural families ofcuts characterizing these representations, and the parameterized linear program itself is used to generate what are usuallydeepest cuts for building up the representations.In this paper, Benders' approach is generalized to a broader class of programs in which the parametrized subproblem need no longer be a linear program. Nonlinear convex duality theory is employed to derive the natural families of cuts corresponding to those in Benders' case. The conditions under which such a generalization is possible and appropriate are examined in detail. An illustrative specialization is made to the variable factor programming problem introduced by R. Wilson, where it offers an especially attractive approach. Preliminary computational experience is given.Communicated by A. V. BalakrishnanAn earlier version was presented at the Nonlinear Programming Symposium at the University of Wisconsin sponsored by the Mathematics Research Center, US Army, May 4–6, 1970. This research was supported by the National Science Foundation under Grant No. GP-8740.     

On the Continuity of the Optimal Value in Parametric Linear Optimization: Stable Discretization of the Lagrangian Dual of Nonlinear Problems

This paper is focused on the stability of the optimal value, and its immediate repercussion on the stability of the optimal set, for a general parametric family of linear optimization problems in ⁿ. In our approach, the parameter ranges over an arbitrary metric space, and each parameter determines directly a set of coefficient vectors describing the linear system of constraints. Thus, systems associated with different parameters are not required to have the same number (cardinality) of inequalities. In this way, discretization techniques for solving a nominal linear semi-infinite optimization problem may be modeled in terms of suitable parametrized problems. The stability results given in the paper are applied to the stability analysis of the Lagrangian dual associated with a parametric family of nonlinear programming problems. This dual problem is translated into a linear (semi-infinite) programming problem and, then, we prove that the lower semicontinuity of the corresponding feasible set mapping, the continuity of the optimal value function, and the upper semicontinuity of the optimal set mapping are satisfied. Then, the paper shows how these stability properties for the dual problem entail a nice behavior of parametric approximation and discretization strategies (in which an ordinary linear programming problem may be considered in each step). This approximation–discretization process is formalized by means of considering a double parameter: the original one and the finite subset of indices (grid) itself. Finally, the convex case is analyzed, showing that the referred process also allows us to approach the primal problem.Mathematics Subject Classifications (2000) Primary 90C34, 90C31; secondary 90C25, 90C05.     

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.     

Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.