Branch-and-bound algorithms for the partial inverse mixed integer linear programming problem

This paper presents branch-and-bound algorithms for the partial inverse mixed integer linear programming (PInvMILP) problem, which is to find a minimal perturbation to the objective function of a mixed integer linear program (MILP), measured by some norm, such that there exists an optimal solution to the perturbed MILP that also satisfies an additional set of linear constraints. This is a new extension to the existing inverse optimization models. Under the weighted $L_1$ and $L_\infty $ norms, the presented algorithms are proved to finitely converge to global optimality. In the presented algorithms, linear programs with complementarity constraints (LPCCs) need to be solved repeatedly as a subroutine, which is analogous to repeatedly solving linear programs for MILPs. Therefore, the computational complexity of the PInvMILP algorithms can be expected to be much worse than that of MILP or LPCC. Computational experiments show that small-sized test instances can be solved within a reasonable time period.     

Dual existence theorems for a special class of linear programming problems in reflexive Banach lattices

In this work the existence of dual optimal solutions for a special class of linear programming problems in a reflexive Banach space is investigated. Then these statements are applied to linear optimization problems with Noethebian operator-constraints. Finally, a maximal condition for an optimal control problem with Noehebiari operator: constraints its Derived in L _∞[0,T].

     

Discontinuous piecewise linear optimization

A theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known. A penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions. Although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions, it is straightforwardly extendable, using standard nonlinear programming techniques, tononlinear (discontinuous piecewise differentiable) functions.The descent algorithm which is elaborated uses active-set and projected gradient approaches. It is a generalization of the ideas used by Conn to deal with nonsmoothness in thel ₁ exact penalty function, and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part. The constrained case is reduced to the unconstrained minimization of a (piecewise linear)l ₁ exact penalty function. We also discuss how the algorithm is modified when it encounters degenerate points. Preliminary numerical results are presented: the algorithm is applied to discontinuous optimization problems from models in industrial engineering. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Supported by the Natural Sciences and Engineering Council of Canada and the Centre de Recherches Mathématiques, Université de Montréal.This research was supported in part by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract No. F49620-91-C-0079. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.     

Minimax linear programming problem

In this paper, we consider the following minimax linear programming problem: min z = max_{1 ≤ j ≤ n}{C_jX_j}, subject to Ax = g, x ≥ 0. It is well known that this problem can be transformed into a linear program by introducing n additional constraints. We note that these additional constraints can be considered implicitly by treating them as parametric upper bounds. Based on this approach we develop two algorithms: a parametric algorithm and a primal—dual algorithm. The parametric algorithm solves a linear programming problem with parametric upper bounds and the primal—dual algorithm solves a sequence of related dual feasible linear programming problems. Computation results are also presented, which indicate that both the algorithms are substantially faster than the simplex algorithm applied to the enlarged linear programming problem.     

A gaussian upper bound for gaussian multi-stage stochastic linear programs

This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to find upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semidefinite program. The algorithm for the multi-stage problem invovles the solution of a quadratically constrained convex programming problem.     

On p-norm linear discrimination

We consider a p-norm linear discrimination model that generalizes the model of Bennett and Mangasarian (1992) and reduces to a linear programming problem with p-order cone constraints. The proposed approach for handling linear programming problems with p-order cone constraints is based on reformulation of p-order cone optimization problems as second order cone programming (SOCP) problems when p is rational. Since such reformulations typically lead to SOCP problems with large numbers of second order cones, an “economical” representation that minimizes the number of second order cones is proposed. A case study illustrating the developed model on several popular data sets is conducted.     

Continuous-time linear programming problems revisited: A perturbation approach

The class of continuous-time linear programming problems under the assumption that the constraints are satisfied almost everywhere in the time interval [0,?T]?is taken into account in this article. Under this assumption, its corresponding discretized problems cannot be formulated by equally dividing the time interval [0,?T]?as subintervals of [0,?T]?. In this article, we also introduce the perturbed continuous-time linear programming problems to prove the strong duality theorem when the constraints are assumed to be satisfied a.e. in [0,?T]?.     

Risk optimization with p-order conic constraints: A linear programming approach

The paper considers solving of linear programming problems with p-order conic constraints that are related to a certain class of stochastic optimization models with risk objective or constraints. The proposed approach is based on construction of polyhedral approximations for p-order cones, and then invoking a Benders decomposition scheme that allows for efficient solving of the approximating problems. The conducted case study of portfolio optimization with p-order conic constraints demonstrates that the developed computational techniques compare favorably against a number of benchmark methods, including second-order conic programming methods.     

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.Part I derives and justifies the essential steps of the algorithm, by extension from the simplex method for linear programming in bounded variables. The proof employs familiar finite-termination arguments and established piecewise-linear duality theory.Part II considers the relaxation of technical assumptions pertaining to finiteness, feasibility and nondegeneracy of piecewise-linear programs. Degeneracy is found to have broader consequences than in the linear case, and the standard techniques for prevention of cycling are extended accordingly.Part III analyzes the computational requirements of piecewise-linear programming. The direct approach embodied in the piecewise-linear simplex algorithm is shown to be inherently more efficient than indirect approaches that rely on transformation of piecewise-linear programs to equivalent linear programs. A concluding section surveys the many applications of piecewise-linear programming in linear programming,l ₁ estimation, goal programming, interval programming, and nonlinear optimization.This research has been supported in part by the National Science Foundation under grant MCS-8217261.     

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.     

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.     

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 algorithm for the estimation of a regression function by continuous piecewise linear functions

The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L ₂ risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.     

Sign consistent linear programming problems

This article studies linear programming problems in which all minors of maximal order of the coefficient matrix have the same sign. We analyse the relationship between a special structure of the non-degenerate dual feasible bases of a linear programming problem and the structure of its associated matrix. In the particular case in which the matrix has all minors of each order k with the same strict sign ? _k , we provide a dual simplex revised method with good stability properties. In particular, this method can be applied to the totally positive linear programming problems, of great interest in many applications.     

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.     

A cone complementarity linearization approach to robust controller design for continuous-time piecewise linear systems with linear fractional uncertainties

This paper investigates the robust H_∞ control problem for uncertain continuous-time piecewise systems by using the piecewise continuous Lyapunov function. The uncertainties of the systems under consideration are expressed in a linear fractional form. A strict linear matrix inequality approach is developed to obtain stability condition and H_∞ performance. The H_∞ controller design problem is solved by exploiting the cone complementarity linearization (CCL) method, which can be cast into an iterative minimization problem subject to LMI constraints. Finally two examples are given to illustrate the application of the proposed approach.     

Solving linear optimization over arithmetic constraint formula

Since Balas extended the classical linear programming problem to the disjunctive programming (DP) problem where the constraints are combinations of both logic AND and OR, many researchers explored this optimization problem under various theoretical or application scenarios such as generalized disjunctive programming (GDP), optimization modulo theories (OMT), robot path planning, real-time systems, etc. However, the possibility of combining these differently-described but form-equivalent problems into a single expression remains overlooked. The contribution of this paper is two folded. First, we convert the linear DP/GDP model, linear-arithmetic OMT problem and related application problems into an equivalent form, referred to as the linear optimization over arithmetic constraint formula (LOACF). Second, a tree-search-based algorithm named RS-LPT is proposed to solve LOACF. RS-LPT exploits the techniques of interval analysis and nonparametric estimation for reducing the search tree and lowering the number of visited nodes. Also, RS-LPT alleviates bad construction of search tree by backtracking and pruning dynamically. We evaluate RS-LPT against two most common DP/GDP methods, three state-of-the-art OMT solvers and the disjunctive transformation based method on optimization benchmarks with different types and scales. Our results favor RS-LPT as compared to existing competing methods, especially for large scale cases.     

Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization

This paper considers Stackelberg solutions for two-level linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random two-level linear programming problem, an α-stochastic two-level linear programming problem is defined through the introduction of α-level sets of fuzzy random variables. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through fractile criterion optimization in stochastic programming, the transformed stochastic two-level programming problem can be reduced to a deterministic two-level programming problem. An extended concept of Stackelberg solution is introduced and a numerical example is provided to illustrate the proposed method.     

A globally convergent method for semi-infinite linear programming

This paper presents a globally convergent method for solving a general semi-infinite linear programming problem. Some important features of this method include: It can solve a semi-infinite linear program having an unbounded feasible region. It requires an inexact solution to a nonlinear subproblem at each iteration. It allows unbounded index sets and nondifferentiable constraints. The amount of work at each iteration k does not increase with k.