Global optimization of mixed-integer bilevel programming problems

Two approaches that solve the mixed-integer nonlinear bilevel programming problem to global optimality are introduced. The first addresses problems mixed-integer nonlinear in outer variables and C²-nonlinear in inner variables. The second adresses problems with general mixed-integer nonlinear functions in outer level. Inner level functions may be mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in inner integer variables, and linear in inner continuous variables. This second approach is based on reformulating the mixed-integer inner problem as continuous via its vertex polyheral convex hull representation and solving the resulting nonlinear bilevel optimization problem by a novel deterministic global optimization framework. Computational studies illustrate proposed approaches.     

Global optimization of bilinear programs with a multiparametric disaggregation technique

In this paper, we present the derivation of the multiparametric disaggregation technique (MDT) by Teles et al. (J. Glob. Optim., 2011) for solving nonconvex bilinear programs. Both upper and lower bounding formulations corresponding to mixed-integer linear programs are derived using disjunctive programming and exact linearizations, and incorporated into two global optimization algorithms that are used to solve bilinear programming problems. The relaxation derived using the MDT is shown to scale much more favorably than the relaxation that relies on piecewise McCormick envelopes, yielding smaller mixed-integer problems and faster solution times for similar optimality gaps. The proposed relaxation also compares well with general global optimization solvers on large problems.     

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.     

The Euclidean Steiner tree problem in Rn: A mathematical programming formulation

A nonconvex mixed-integer programming formulation for the Euclidean Steiner Tree Problem (ESTP) in Rⁿ is presented. After obtaining separability between integer and continuous variables in the objective function, a Lagrange dual program is proposed. To solve this dual problem (and obtaining a lower bound for ESTP) we use subgradient techniques. In order to evaluate a subgradient at each iteration we have to solve three optimization problems, two in polynomial time, and one is a special convex nondifferentiable programming problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

Global optimization of signomial mixed-integer nonlinear programming problems with free variables

Mixed-integer nonlinear programming (MINLP) problems involving general constraints and objective functions with continuous and integer variables occur frequently in engineering design, chemical process industry and management. Although many optimization approaches have been developed for MINLP problems, these methods can only handle signomial terms with positive variables or find a local solution. Therefore, this study proposes a novel method for solving a signomial MINLP problem with free variables to obtain a global optimal solution. The signomial MINLP problem is first transformed into another one containing only positive variables. Then the transformed problem is reformulated as a convex mixed-integer program by the convexification strategies and piecewise linearization techniques. A global optimum of the signomial MINLP problem can finally be found within the tolerable error. Numerical examples are also presented to demonstrate the effectiveness of the proposed method.     

Logic-Based Modeling and Solution of Nonlinear Discrete/Continuous Optimization Problems

This paper presents a review of advances in the mathematical programming approach to discrete/continuous optimization problems. We first present a brief review of MILP and MINLP for the case when these problems are modeled with algebraic equations and inequalities. Since algebraic representations have some limitations such as difficulty of formulation and numerical singularities for the nonlinear case, we consider logic-based modeling as an alternative approach, particularly Generalized Disjunctive Programming (GDP), which the authors have extensively investigated over the last few years. Solution strategies for GDP models are reviewed, including the continuous relaxation of the disjunctive constraints. Also, we briefly review a hybrid model that integrates disjunctive programming and mixed-integer programming. Finally, the global optimization of nonconvex GDP problems is discussed through a two-level branch and bound procedure.     

Conic mixed-integer rounding cuts

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability. This research has been supported, in part, by Grant # DMI0700203 from the National Science Foundation.     

Mixed-integer bilinear programming problems

This paper addresses a class of problems called mixed-integer bilinear programming problems. These problems are identical to the well known bilinear programming problems with the exception that one set of variables is restricted to be binary valued, and they arise in various production, location—allocation, and distribution application contexts. We first identify some special cases of this problem which are relatively more readily solvable, even though their continuous relaxations are still nonconvex. For the more general case, we employ a linearization technique and design a composite Lagrangian relaxation-implicit enumeration-cutting plane algorithm. Extensive computational experience is provided to test the efficacy of various algorithmic strategies and the effects of problem data on the computational effort of the proposed algorithm.     

Multi-parametric disaggregation technique for global optimization of polynomial programming problems

This paper discusses a power-based transformation technique that is especially useful when solving polynomial optimization problems, frequently occurring in science and engineering. The polynomial nonlinear problem is primarily transformed into a suitable reformulated problem containing new sets of discrete and continuous variables. By applying a term-wise disaggregation scheme combined with multi-parametric elements, an upper/lower bounding mixed-integer linear program can be derived for minimization/maximization problems. It can then be solved to global optimality through standard methods, with the original problem being approximated to a certain precision level, which can be as tight as desired. Furthermore, this technique can also be applied to signomial problems with rational exponents, after a few effortless algebraic transformations. Numerical examples taken from the literature are used to illustrate the effectiveness of the proposed approach.     

A trust region SQP algorithm for mixed-integer nonlinear programming

We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.     

An Improved Bernstein Global Optimization Algorithm for MINLP Problems with Application in Process Industry

We present an improved Bernstein global optimization algorithm to solve polynomial mixed-integer nonlinear programming (MINLP) problems. The algorithm is of branch-and-bound type, and uses the Bernstein form of the polynomials for the global optimization. The new ingredients in the algorithm include a modified subdivision procedure, a vectorized Bernstein cut-off test and a new branching rule for the decision variables. The performance of the improved algorithm is tested and compared with earlier reported Bernstein global optimization algorithm (to solve polynomial MINLPs) and with several state-of-the-art MINLP solvers on a set of 19 test problems. The results of the tests show the superiority of the improved algorithm over the earlier reported Bernstein algorithm and the state-of-the-art solvers in terms of the chosen performance metrics. Similarly, efficacy of the improved algorithm in handling a real-world MINLP problem is brought out via a trim-loss minimization problem from the process industry.     

A branch-and-reduce approach to global optimization

This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.     

Generation of disjointly constrained bilinear programming test problems

This paper describes a technique for generating disjointly constrained bilinear programming test problems with known solutions and properties. The proposed construction technique applies a simple random transformation of variables to a separable bilinear programming problem that is constructed by combining disjoint low-dimensional bilinear programs.     

An evolutionary programming approach to mixed-variable optimization problems

Many engineering optimization problems frequently encounter discrete variables as well as continuous variables and the presence of nonlinear discrete variables considerably adds to the solution complexity. Very few of the existing methods can find a globally optimal solution when the objective functions are non-convex and non-differentiable. In this paper, we present a mixed-variable evolutionary programming (MVEP) technique for solving these nonlinear optimization problems which contain integer, discrete, zero-one and continuous variables. The MVEP provides an improvement in global search reliability in a mixed-variable space and converges steadily to a good solution. An approach to handle various kinds of variables and constraints is discussed. Some examples of mixed-variable optimization problems in the literature are tested, which demonstrate that the proposed approach is superior to current methods for finding the best solution, in terms of both solution quality and algorithm robustness.     

Optimizing fuzzy p-hub center problem with generalized value-at-risk criterion

In this study, we reduce the uncertainty embedded in secondary possibility distribution of a type-2 fuzzy variable by fuzzy integral, and apply the proposed reduction method to p-hub center problem, which is a nonlinear optimization problem due to the existence of integer decision variables. In order to optimize p-hub center problem, this paper develops a robust optimization method to describe travel times by employing parametric possibility distributions. We first derive the parametric possibility distributions of reduced fuzzy variables. After that, we apply the reduction methods to p-hub center problem and develop a new generalized value-at-risk (VaR) p-hub center problem, in which the travel times are characterized by parametric possibility distributions. Under mild assumptions, we turn the original fuzzy p-hub center problem into its equivalent parametric mixed-integer programming problems. So, we can solve the equivalent parametric mixed-integer programming problems by general-purpose optimization software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the efficiency of the proposed solution methods.     

Numerical solution of bilinear programming problems

A bilinear programming problem with uncoupled variables is considered. First, a special technique for generating test bilinear problems is considered. Approximate algorithms for local and global search are proposed. Asymptotic convergence of these algorithms is analyzed, and stopping rules are proposed. In conclusion, numerical results for randomly generated bilinear problems are presented and analyzed.     

Solving air traffic conflict problems via local continuous optimization

This paper first introduces an original trajectory model using B-splines and a new semi-infinite programming formulation of the separation constraint involved in air traffic conflict problems. A new continuous optimization formulation of the tactical conflict-resolution problem is then proposed. It involves very few optimization variables in that one needs only one optimization variable to determine each aircraft trajectory. Encouraging numerical experiments show that this approach is viable on realistic test problems. Not only does one not need to rely on the traditional, discretized, combinatorial optimization approaches to this problem, but, moreover, local continuous optimization methods, which require relatively fewer iterations and thereby fewer costly function evaluations, are shown to improve the performance of the overall global optimization of this non-convex problem.     

Strong valid inequalities for orthogonal disjunctions and bilinear covering sets

In this paper, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when this convex hull is completely determined by orthogonal restrictions of the original set. Although the tools used in this construction include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and finding the convex hull of certain mixed/pure-integer bilinear sets. We then extend the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the non-negative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant. Although we illustrate our results primarily on bilinear covering sets, they also apply to more general polynomial covering sets for which they yield new tight relaxations.     

Global optimization of general nonconvex problems with intermediate polynomial substructures

This work considers the global optimization of general nonconvex nonlinear and mixed-integer nonlinear programming problems with underlying polynomial substructures. We incorporate linear cutting planes inspired by reformulation-linearization techniques to produce tight subproblem formulations that exploit these underlying structures. These cutting plane strategies simultaneously convexify linear and nonlinear terms from multiple constraints and are highly effective at tightening standard linear programming relaxations generated by sequential factorable programming techniques. Because the number of available cutting planes increases exponentially with the number of variables, we implement cut filtering and selection strategies to prevent an exponential increase in relaxation size. We introduce algorithms for polynomial substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in BARON at every node in the branch-and-bound tree. A computational study including randomly generated problems of varying size and complexity demonstrates that the exploitation of underlying polynomial substructures significantly reduces computational time, branch-and-bound tree size, and required memory.     

A new technique for generating quadratic programming test problems

This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated.Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems.Support of this work has been provided by the Instituto Nacional de Investigação Científica de Portugal (INIC) under contract 89/EXA/5 and by the Natural Sciences and Engineering Research Council of Canada operating grant 5671.Much of this paper was completed while this author was on a research sabbatical at the Universidade de Coimbra, Portugal.