Enumeration Approach for Linear Complementarity Problems Based on a Reformulation-Linearization Technique

In this paper, we consider the linear complementarity problem (LCP) and present a global optimization algorithm based on an application of the reformulation-linearization technique (RLT). The matrix M associated with the LCP is not assumed to possess any special structure. In this approach, the LCP is formulated first as a mixed-integer 0–1 bilinear programming problem. The RLT scheme is then used to derive a new equivalent mixed-integer linear programming formulation of the LCP. An implicit enumeration scheme is developed that uses Lagrangian relaxation, strongest surrogate and strengthened cutting planes, and a heuristic, designed to exploit the strength of the resulting linearization. Computational experience on various test problems is presented.     

Dynamic Lagrangian dual and reduced RLT constructs for solving 0–1 mixed-integer programs

In this paper, we consider a dynamic Lagrangian dual optimization procedure for solving mixed-integer 0–1 linear programming problems. Similarly to delayed relax-and-cut approaches, the procedure dynamically appends valid inequalities to the linear programming relaxation as induced by the Reformulation-Linearization Technique (RLT). A Lagrangian dual algorithm that is augmented with a primal solution recovery scheme is applied implicitly to a full or partial first-level RLT relaxation, where RLT constraints that are currently being violated by the primal estimate are dynamically generated within the Lagrangian dual problem, thus controlling the size of the dual space while effectively capturing the strength of the RLT-enhanced relaxation. We present a preliminary computational study to demonstrate the efficacy of this approach.     

A Reformulation-Linearization Technique (RLT) for semi-infinite and convex programs under mixed 0-1 and general discrete restrictions

The Reformulation-Linearization Technique (RLT) provides a hierarchy of relaxations spanning the spectrum from the continuous relaxation to the convex hull representation for linear 0-1 mixed-integer and general mixed-discrete programs. We show in this paper that this result holds identically for semi-infinite programs of this type. As a consequence, we extend the RLT methodology to describe a construct for generating a hierarchy of relaxations leading to the convex hull representation for bounded 0-1 mixed-integer and general mixed-discrete convex programs, using an equivalent semi-infinite linearized representation for such problems as an intermediate stepping stone in the analysis. For particular use in practice, we provide specialized forms of the resulting first-level RLT formulation for such mixed 0-1 and discrete convex programs, and illustrate these forms through two examples.     

DRL*: A hierarchy of strong block-decomposable linear relaxations for 0-1 MIPs

In this paper we introduce DRL*, a new hierarchy of linear relaxations for 0-1 mixed integer linear programs (MIPs), based on the idea of Reformulation-Linearization, and explore its links with the Lift-and-Project (L&P) hierarchy and the Sherali-Adams (RLT) hierarchy. The relaxations of the new hierarchy are shown to be intermediate in strength between L&P and RLT relaxations, and examples are shown for which it leads to significantly stronger bounds than those obtained from Lift-and-Project relaxations. On the other hand, as opposed to the RLT relaxations, a key advantage of the DRL* relaxations is that they feature a decomposable structure when formulated in extended space, therefore lending themselves to more efficient solution algorithms by properly exploiting decomposition. Links between DRL* and both the L&P and RLT hierarchies are further explored, and those constraints which should be added to the rank d L&P relaxation (resp to the rank d RLT relaxation) to make it coincide with the rank d DRL* relaxation (resp: to the rank d RLT relaxation) are identified. Furthermore, a full characterization of those 0-1 MIPs for which the DRL* and RLT relaxations coincide is obtained. As an application, we show that both the RLT and DRL* relaxations are the same up to rank d for the problem of optimizing a pseudoboolean function of degree d over a polyhedron. We report computational results comparing the strengths of the rank 2 L&P, DRL* and RLT relaxations. Impact on possible improved efficiency in computing some bounds for the quadratic assignment problem and other directions for future research are suggested in the conclusions.     

Stochastic programming with integer variables

 Including integer variables into traditional stochastic linear programs has considerable implications for structural analysis and algorithm design. Starting from mean-risk approaches with different risk measures we identify corresponding two- and multi-stage stochastic integer programs that are large-scale block-structured mixed-integer linear programs if the underlying probability distributions are discrete. We highlight the role of mixed-integer value functions for structure and stability of stochastic integer programs. When applied to the block structures in stochastic integer programming, well known algorithmic principles such as branch-and-bound, Lagrangian relaxation, or cutting plane methods open up new directions of research. We review existing results in the field and indicate departure points for their extension. Received: December 2, 2002 / Accepted: April 23, 2003 Published online: May 28, 2003 Mathematics Subject Classification (2000): 90C15, 90C11, 90C06, 90C57     

Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y _ij to represent each of the products x _i x _j of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y - x x^T \succeq 0{Y - x x^T \succeq 0} and disjunctive programming. On the other hand, the main drawback of such an extended formulation is its huge size, even for problems for which the number of x _i variables is moderate. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen-reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on box-QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 s, even for large instances with 100 variables; the SDP+RLT relaxations for the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver.     

Tightening concise linear reformulations of 0-1 cubic programs

A common strategy for solving 0-1 cubic programs is to reformulate the non-linear problem into an equivalent linear representation, which can then be submitted directly to a standard mixed-integer programming solver. Both the size and the strength of the continuous relaxation of the reformulation determine the success of this method. One of the most compact linear representations of 0-1 cubic programs is based on a repeated application of the linearization technique for 0-1 quadratic programs introduced by Glover. In this paper, we develop a pre-processing step that serves to strengthen the linear programming bound provided by this concise linear form of a 0-1 cubic program. The proposed scheme involves using optimal dual multipliers of a partial level-2 RLT formulation to rewrite the objective function of the cubic program before applying the linearization. We perform extensive computational tests on the 0-1 cubic multidimensional knapsack problem to show the advantage of our approach.     

Rounding-based heuristics for nonconvex MINLPs

We propose two primal heuristics for nonconvex mixed-integer nonlinear programs. Both are based on the idea of rounding the solution of a continuous nonlinear program subject to linear constraints. Each rounding step is accomplished through the solution of a mixed-integer linear program. Our heuristics use the same algorithmic scheme, but they differ in the choice of the point to be rounded (which is feasible for nonlinear constraints but possibly fractional) and in the linear constraints. We propose a feasibility heuristic, that aims at finding an initial feasible solution, and an improvement heuristic, whose purpose is to search for an improved solution within the neighborhood of a given point. The neighborhood is defined through local branching cuts or box constraints. Computational results show the effectiveness in practice of these simple ideas, implemented within an open-source solver for nonconvex mixed-integer nonlinear programs.     

Mixing mixed-integer inequalities

Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure. Received: April 1998 / Accepted: January 2001?Published online April 12, 2001     

Reduced RLT representations for nonconvex polynomial programming problems

This paper explores equivalent, reduced size Reformulation-Linearization Technique (RLT)-based formulations for polynomial programming problems. Utilizing a basis partitioning scheme for an embedded linear equality subsystem, we show that a strict subset of RLT defining equalities imply the remaining ones. Applying this result, we derive significantly reduced RLT representations and develop certain coherent associated branching rules that assure convergence to a global optimum, along with static as well as dynamic basis selection strategies to implement the proposed procedure. In addition, we enhance the RLT relaxations with v-semidefinite cuts, which are empirically shown to further improve the relative performance of the reduced RLT method over the usual RLT approach. We present computational results for randomly generated instances to test the different proposed reduction strategies and to demonstrate the improvement in overall computational effort when such reduced RLT mechanisms are employed.     

On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables

In this paper, we propose a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having mixed-integer first- and second-stage variables. A modified Benders' decomposition method is developed, where the Benders' subproblems define lower bounding second-stage value functions of the first-stage variables that are derived by constructing a certain partial convex hull representation of the two-stage solution space. This partial convex hull is sequentially generated using a convexification scheme such as the Reformulation-Linearization Technique (RLT) or lift-and-project process, which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first-stage variables. A branch-and-bound algorithm is designed based on a hyperrectangular partitioning process, using the established property that any resulting lower bounding Benders' master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first-stage variable solution is an extreme point of the defining hyperrectangle or the second-stage solution satisfies the binary restrictions. We prove that this algorithm converges to a global optimal solution. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach.     

A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems

Given a set ofR ⁿ and a functionf from intoR ⁿ we consider a problem of finding a pointx ^* in such that(x–x ^*) ^t f(x ^*)0 holds for every pointx in. This problem is called the stationary point problem and the pointx ^* is called a stationary point. We present a variable dimension algorithm for solving the stationary point problem with an affine functionf on a polytope defined by constraints of linear equations and inequalities. We propose a system of equations whose solution set contains a piecewise linear path connecting a trivial starting point in with a stationary point. The path can be followed by solving a series of linear programs which inherit the structure of constraints of. The linear programs are solved efficiently with the Dantzig-Wolfe decomposition method by exploiting fully the structure.Part of this research was carried out when the first author was supported by the Center for Economic Research, Tilburg University, The Netherlands and the third author was supported by the Alexander von Humboldt-Foundation, Federal Republic of Germany.     

Safe bounds in linear and mixed-integer linear programming

Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many state-of-the-art MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size. Mathematics Subject Classification (2000):primary 90C11, secondary 65G20     

On a Dulac function for the Kukles system

To estimate the number and location of limit cycles of Kukles systems in a strip of the phase plane xOy, we develop a method for constructing the Dulac function in the form of a polynomial in the phase variable y with coefficients depending on the second phase variable x. The suggested method is regular for h ₂(x) = 0. The proof of the fact that the constructed function is a Dulac function is reduced to finding a specific number of positive functions that are linear combinations of known functions of the single phase variable x with arbitrary constants. To construct the Dulac function, we use a solution of the corresponding linear programming problem. In addition, we show that the presented approach is efficient from the practical viewpoint and permits one to obtain a global exact estimate for the number of limit cycles of above-mentioned systems in some cases.     

Branch-and-price for a class of nonconvex mixed-integer nonlinear programs

This work attempts to combine the strengths of two major technologies that have matured over the last three decades: global mixed-integer nonlinear optimization and branch-and-price. We consider a class of generally nonconvex mixed-integer nonlinear programs (MINLPs) with linear complicating constraints and integer linking variables. If the complicating constraints are removed, the problem becomes easy to solve, e.g. due to decomposable structure. Integrality of the linking variables allows us to apply a discretization approach to derive a Dantzig-Wolfe reformulation and solve the problem to global optimality using branch-andprice. It is a remarkably simple idea; but to our surprise, it has barely found any application in the literature. In this work, we show that many relevant problems directly fall or can be reformulated into this class of MINLPs. We present the branch-and-price algorithm and demonstrate its effectiveness (and sometimes ineffectiveness) in an extensive computational study considering multiple large-scale problems of practical relevance, showing that, in many cases, orders-of-magnitude reductions in solution time can be achieved.

     

Piecewise-Linear Approximations of Multidimensional Functions

We develop explicit, piecewise-linear formulations of functions f(x):ℝ ⁿ ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.     

Depth-Optimized Convexity Cuts

This paper presents a general, self-contained treatment of convexity or intersection cuts. It describes two equivalent ways of generating a cut—via a convex set or a concave function—and a partial-order notion of cut strength. We then characterize the structure of the sets and functions that generate cuts that are strongest with respect to the partial order. Next, we specialize this analytical framework to the case of mixed-integer linear programming (MIP). For this case, we formulate two kinds of the deepest cut generation problem, via sets or via functions, and subsequently consider some special cases which are amenable to efficient computation. We conclude with computational tests of one of these procedures on a large set of MIPLIB problems.     

Combined bound-grid-factor constraints for enhancing RLT relaxations for polynomial programs

This paper studies the global optimization of polynomial programming problems using Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations. We introduce a new class of bound-grid-factor constraints that can be judiciously used to augment the basic RLT relaxations in order to improve the quality of lower bounds and enhance the performance of global branch-and-bound algorithms. Certain theoretical properties are established that shed light on the effect of these valid inequalities in driving the discrepancies between RLT variables and their associated nonlinear products to zero. To preserve computational expediency while promoting efficiency, we propose certain concurrent and sequential cut generation routines and various grid-factor selection rules. The results indicate a significant tightening of lower bounds, which yields an overall reduction in computational effort for solving a test-bed of polynomial programming problems to global optimality in comparison with the basic RLT procedure as well as the commercial software BARON.     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000