Global solution of bilevel programs with a nonconvex inner program

A bounding algorithm for the global solution of nonlinear bilevel programs involving nonconvex functions in both the inner and outer programs is presented. The algorithm is rigorous and terminates finitely to a point that satisfies ε-optimality in the inner and outer programs. For the lower bounding problem, a relaxed program, containing the constraints of the inner and outer programs augmented by a parametric upper bound to the parametric optimal solution function of the inner program, is solved to global optimality. The optional upper bounding problem is based on probing the solution obtained by the lower bounding procedure. For the case that the inner program satisfies a constraint qualification, an algorithmic heuristic for tighter lower bounds is presented based on the KKT necessary conditions of the inner program. The algorithm is extended to include branching, which is not required for convergence but has potential advantages. Two branching heuristics are described and analyzed. Convergence proofs are provided and numerical results for original test problems and for literature examples are presented.     

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.     

On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification

This paper focuses on bilevel programs with a convex lower-level problem violating Slater’s constraint qualification. We relax the constrained domain of the lower-level problem. Then, an approximate solution of the original bilevel program can be obtained by solving this perturbed bilevel program. As the lower-level problem of the perturbed bilevel program satisfies Slater’s constraint qualification, it can be reformulated as a mathematical program with complementarity constraints which can be solved by standard algorithms. The lower convergence properties of the constraint set mapping and the solution set mapping of the lower-level problem of the perturbed bilevel program are expanded. We show that the solutions of a sequence of the perturbed bilevel programs are convergent to that of the original bilevel program under some appropriate conditions. And this convergence result is applied to simple trilevel programs.     

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.      

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.     

Designing robust coverage networks to hedge against worst-case facility losses

In order to design a coverage-type service network that is robust to the worst instances of long-term facility loss, we develop a facility location–interdiction model that maximizes a combination of initial coverage by p facilities and the minimum coverage level following the loss of the most critical r facilities. The problem is formulated both as a mixed-integer program and as a bilevel mixed-integer program. To solve the bilevel program optimally, a decomposition algorithm is presented, whereby the original bilevel program is decoupled into an upper level master problem and a lower level subproblem. After sequentially solving these problems, supervalid inequalities can be generated and appended to the upper level master in an attempt to force it away from clearly dominated solutions. Computational results show that when solved to optimality, the bilevel decomposition algorithm is up to several orders of magnitude faster than performing branch and bound on the mixed-integer program.     

A Benders decomposition method for discretely-constrained mathematical programs with equilibrium constraints

We present a new methodology to solve discretely-constrained mathematical programs with equilibrium constraints (DC-MPECs). Typically these problems include an upper planning-level optimization with some discrete decision variables (eg, build/don’t build) as well as a lower operations-level problem often described by an optimization or nonlinear complementarity problem. This lower-level problem may also include some discrete variables. MPECs are very challenging problems to solve and the inclusion of integrality constraints makes this class of problems even more computationally difficult. We develop a new variant of the Benders algorithm combined with a heuristic procedure that decomposes the domain of the upper-level discrete variables to solve the resulting DC-MPECs. We provide convergence theory as well as a number of numerical examples, some derived from energy applications, to validate the new method. It should be noted that the convergence theory applies if the heuristic procedure correctly identifies a decomposition of the domain so that the lower-level problem's optimal value function is convex. This is challenging but our numerical results are positive.     

Parametric mixed-integer 0–1 linear programming: The general case for a single parameter

Two algorithms for the general case of parametric mixed-integer linear programs (MILPs) are proposed. Parametric MILPs are considered in which a single parameter can simultaneously influence the objective function, the right-hand side and the matrix. The first algorithm is based on branch-and-bound on the integer variables, solving a parametric linear program (LP) at each node. The second algorithm is based on the optimality range of a qualitatively invariant solution, decomposing the parametric optimization problem into a series of regular MILPs, parametric LPs and regular mixed-integer nonlinear programs (MINLPs). The number of subproblems required for a particular instance is equal to the number of critical regions. For the parametric LPs an improvement of the well-known rational simplex algorithm is presented, that requires less consecutive operations on rational functions. Also, an alternative based on predictor–corrector continuation is proposed. Numerical results for a test set are discussed.     

A Trust-Region Method for Nonlinear Bilevel Programming: Algorithm and Computational Experience

We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upper-level objective and all constraint-defining functions, as well as a quadratic approximation of the lower-level objective. We describe the main features of the algorithm and the resulting software. Numerical experiments tend to confirm the promising behavior of the method.     

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.     

A single-level reformulation of mixed integer bilevel programming problems

This paper focuses on the single-level reformulation of mixed integer bilevel programming problems (MIBLPP). Due to the existence of lower-level integer variables, the popular approaches in the literature such as the first-order approach are not applicable to the MIBLPP. In this paper, we reformulate the MIBLPP as a mixed integer mathematical program with complementarity constraints (MIMPCC) by separating the lower-level continuous and integer variables. In particular, we show that global and local minimizers of the MIBLPP correspond to those of the MIMPCC respectively under suitable conditions.     

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.     

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader’s variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case, it yields a “better than fully polynomial time” approximation scheme with running time polynomial in the logarithm of the absolute precision. For the pure integer case where the leader’s variables are integer, and hence optimal solutions are guaranteed to exist, we present an algorithm which runs in polynomial time when the total number of variables is fixed.     

A branch-and-bound method for discretely-constrained mathematical programs with equilibrium constraints

We present a branch-and-bound algorithm for discretely-constrained mathematical programs with equilibrium constraints (DC-MPEC). This is a class of bilevel programs with an integer program in the upper-level and a complementarity problem in the lower-level. The algorithm builds on the work by Gabriel et al. (Journal of the Operational Research Society 61(9):1404–1419, 2010) and uses Benders decomposition to form a master problem and a subproblem. The new dynamic partition scheme that we present ensures that the algorithm converges to the global optimum. Partitioning is done to overcome the non-convexity of the Benders subproblem. In addition Lagrangean relaxation provides bounds that enable fathoming in the branching tree and warm-starting the Benders algorithm. Numerical tests show significantly reduced solution times compared to the original algorithm. When the lower level problem is stochastic our algorithm can easily be further decomposed using scenario decomposition. This is demonstrated on a realistic case.     

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.     

Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints

This paper studies a bilevel polynomial program involving box data uncertainties in both its linear constraint set and its lower-level optimization problem. We show that the robust global optimal value of the uncertain bilevel polynomial program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. This is done by first transforming the uncertain bilevel polynomial program into a single-level non-convex polynomial program using a dual characterization of the solution of the lower-level program and then employing the powerful Putinar’s Positivstellensatz of semi-algebraic geometry. We provide a numerical example to show how the robust global optimal value of the uncertain bilevel polynomial program can be calculated by solving a semidefinite programming problem using the MATLAB toolbox YALMIP.     

Mixed-integer quadratic programming

This paper considers mixed-integer quadratic programs in which the objective function is quadratic in the integer and in the continuous variables, and the constraints are linear in the variables of both types. The generalized Benders' decomposition is a suitable approach for solving such programs. However, the program does not become more tractable if this method is used, since Benders' cuts are quadratic in the integer variables. A new equivalent formulation that renders the program tractable is developed, under which the dual objective function is linear in the integer variables and the dual constraint set is independent of these variables. Benders' cuts that are derived from the new formulation are linear in the integer variables, and the original problem is decomposed into a series of integer linear master problems and standard quadratic subproblems. The new formulation does not introduce new primary variables or new constraints into the computational steps of the decomposition algorithm.The author wishes to thank two anonymous referees for their helpful comments and suggestions for revising the paper.     

Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.     

优化方法解决无上层约束的线性双层规划问题

In this paper we propose an optimal method for solving the linear bilevel programming problem with no upper-level constraint. The main idea of this method is that the initial point which is in the feasible region goes forward along the optimal direction firstly. When the iterative point reaches the boundary of the feasible region, it can continue to go forward along the suboptimal direction. The iteration is terminated until the iterative point cannot go forward along the suboptimal direction and effective direction, and the new iterative point is the solution of the lower-level programming. An algorithm which bases on the main idea above is presented and the solution obtained via this algorithm is proved to be optimal solution to the bilevel programming problem. This optimal method is effective for solving the linear bilevel programming problem.     

关于二层规划的逼近问题

下层问题以上层决策变量作为参数,而上层是以下层问题的最优值作为响应 的一类最优化问题——二层规划问题。我们给出了由一系列此类二层规划去逼近原二层规划的逼近法,得到了这种逼近的一些有趣的结果.