Undercover: a primal MINLP heuristic exploring a largest sub-MIP

We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programs (MINLPs) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a smallest set of variables to fix, a so-called cover, such that each constraint is linearized. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. We apply domain propagation to try to avoid infeasibilities, and conflict analysis to learn additional constraints from infeasibilities that are nonetheless encountered. We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and MINLPs. It turns out that the majority of these instances allows for small covers. Although general in nature, we show that the heuristic is most successful on MIQCPs. It nicely complements existing root-node heuristics in different state-of-the-art solvers and helps to significantly improve the overall performance of the MINLP solver SCIP.     

Linearization-based algorithms for mixed-integer nonlinear programs with convex continuous relaxation

We present two linearization-based algorithms for mixed-integer nonlinear programs (MINLPs) having a convex continuous relaxation. The key feature of these algorithms is that, in contrast to most existing linearization-based algorithms for convex MINLPs, they do not require the continuous relaxation to be defined by convex nonlinear functions. For example, these algorithms can solve to global optimality MINLPs with constraints defined by quasiconvex functions. The first algorithm is a slightly modified version of the LP/NLP-based branch-and-bouund \((\text{ LP/NLP-BB })\) algorithm of Quesada and Grossmann, and is closely related to an algorithm recently proposed by Bonami et al. (Math Program 119:331–352, 2009). The second algorithm is a hybrid between this algorithm and nonlinear programming based branch-and-bound. Computational experiments indicate that the modified LP/NLP-BB method has comparable performance to LP/NLP-BB on instances defined by convex functions. Thus, this algorithm has the potential to solve a wider class of MINLP instances without sacrificing performance.     

Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs

This paper considers deterministic global optimization of scenario-based, two-stage stochastic mixed-integer nonlinear programs (MINLPs) in which the participating functions are nonconvex and separable in integer and continuous variables. A novel decomposition method based on generalized Benders decomposition, named nonconvex generalized Benders decomposition (NGBD), is developed to obtain ??-optimal solutions of the stochastic MINLPs of interest in finite time. The dramatic computational advantage of NGBD over state-of-the-art global optimizers is demonstrated through the computational study of several engineering problems, where a problem with almost 150,000 variables is solved by NGBD within 80 minutes of solver time.     

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.     

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.     

An efficient strategy for the activation of MIP relaxations in a multicore global MINLP solver

Solving mixed-integer nonlinear programming (MINLP) problems to optimality is a NP-hard problem, for which many deterministic global optimization algorithms and solvers have been recently developed. MINLPs can be relaxed in various ways, including via mixed-integer linear programming (MIP), nonlinear programming, and linear programming. There is a tradeoff between the quality of the bounds and CPU time requirements of these relaxations. Unfortunately, these tradeoffs are problem-dependent and cannot be predicted beforehand. This paper proposes a new dynamic strategy for activating and deactivating MIP relaxations in various stages of a branch-and-bound algorithm. The primary contribution of the proposed strategy is that it does not use meta-parameters, thus avoiding parameter tuning. Additionally, this paper proposes a strategy that capitalizes on the availability of parallel MIP solver technology to exploit multicore computing hardware while solving MINLPs. Computational tests for various benchmark libraries reveal that our MIP activation strategy works efficiently in single-core and multicore environments.     

Valid inequalities for separable concave constraints with indicator variables

We study valid inequalities for optimization models that contain both binary indicator variables and separable concave constraints. These models reduce to a mixed-integer linear program (MILP) when the concave constraints are ignored, or to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms designed to solve these problems to global optimality, cutting planes to strengthen the relaxation are traditionally obtained using valid inequalities for the MILP only. We propose a technique to obtain valid inequalities that are based on both the MILP constraints and the concave constraints. We begin by characterizing the convex hull of a four-dimensional set consisting of a single binary indicator variable, a single concave constraint, and two linear inequalities. Using this analysis, we demonstrate how valid inequalities for the single node flow set and for the lot-sizing polyhedron can be “tilted” to give valid inequalities that also account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities for nonlinear transportation problems and for lot-sizing problems with concave costs. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed-integer nonlinear programs.     

A global MINLP approach to symbolic regression

Symbolic regression methods generate expression trees that simultaneously define the functional form of a regression model and the regression parameter values. As a result, the regression problem can search many nonlinear functional forms using only the specification of simple mathematical operators such as addition, subtraction, multiplication, and division, among others. Currently, state-of-the-art symbolic regression methods leverage genetic algorithms and adaptive programming techniques. Genetic algorithms lack optimality certifications and are typically stochastic in nature. In contrast, we propose an optimization formulation for the rigorous deterministic optimization of the symbolic regression problem. We present a mixed-integer nonlinear programming (MINLP) formulation to solve the symbolic regression problem as well as several alternative models to eliminate redundancies and symmetries. We demonstrate this symbolic regression technique using an array of experiments based upon literature instances. We then use a set of 24 MINLPs from symbolic regression to compare the performance of five local and five global MINLP solvers. Finally, we use larger instances to demonstrate that a portfolio of models provides an effective solution mechanism for problems of the size typically addressed in the symbolic regression literature.     

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.     

On generalized surrogate duality in mixed-integer nonlinear programming

Mathematical Programming - The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global $$\epsilon $$ -optimality with spatial branch and bound is a tight,...     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

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.     

Univariate parameterization for global optimization of mixed-integer polynomial problems

This paper presents a new relaxation technique to globally optimize mixed-integer polynomial programming problems that arise in many engineering and management contexts. Using a bilinear term as the basic building block, the underlying idea involves the discretization of one of the variables up to a chosen accuracy level (Teles, J.P., Castro, P.M., Matos, H.A. (2013). Multiparametric disaggregation technique for global optimization of polynomial programming problems. J. Glob. Optim. 55, 227–251), by means of a radix-based numeric representation system, coupled with a residual variable to effectively make its domain continuous. Binary variables are added to the formulation to choose the appropriate digit for each position together with new sets of continuous variables and constraints leading to the transformation of the original mixed-integer non-linear problem into a larger one of the mixed-integer linear programming type. The new underestimation approach can be made as tight as desired and is shown capable of providing considerably better lower bounds than a widely used global optimization solver for a specific class of design problems involving bilinear terms.     

LaGO: a (heuristic) Branch and Cut algorithm for nonconvex MINLPs

We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixed-integer nonlinear programs (MINLPs). A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation techniques for the construction of the convex relaxation cannot be applied, and we are restricted to sampling techniques in case of nonquadratic nonconvex functions. The linear relaxation is further improved by mixed-integer-rounding cuts. Also box reduction techniques are applied to improve efficiency. Numerical results on medium size test problems are presented to show the efficiency of the method.     

Solving mixed integer nonlinear programs by outer approximation

A wide range of optimization problems arising from engineering applications can be formulated as Mixed Integer NonLinear Programming problems (MINLPs). Duran and Grossmann (1986) suggest an outer approximation scheme for solving a class of MINLPs that are linear in the integer variables by a finite sequence of relaxed MILP master programs and NLP subproblems.Their idea is generalized by treating nonlinearities in the integer variables directly, which allows a much wider class of problem to be tackled, including the case of pure INLPs. A new and more simple proof of finite termination is given and a rigorous treatment of infeasible NLP subproblems is presented which includes all the common methods for resolving infeasibility in Phase I.The worst case performance of the outer approximation algorithm is investigated and an example is given for which it visits all integer assignments. This behaviour leads us to include curvature information into the relaxed MILP master problem, giving rise to a new quadratic outer approximation algorithm.An alternative approach is considered to the difficulties caused by infeasibility in outer approximation, in which exact penalty functions are used to solve the NLP subproblems. It is possible to develop the theory in an elegant way for a large class of nonsmooth MINLPs based on the use of convex composite functions and subdifferentials, although an interpretation for thel ₁ norm is also given.This work is supported by SERC grant no. SERC GR/F 07972.Corresponding author.     

Energy-Optimal Multi-Goal Motion Planning for Planar Robot Manipulators

In this work, the energy-optimal motion planning problem for planar robot manipulators with two revolute joints is studied, in which the end-effector of the robot manipulator is constrained to pass through a set of waypoints, whose sequence is not predefined. This multi-goal motion planning problem has been solved as a mixed-integer optimal control problem in which, given the dynamic model of the robot manipulator, the initial and final configurations of the robot, and a set of waypoints inside the workspace of the manipulator, one has to find the control inputs, the sequence of waypoints with the corresponding passage times, and the resulting trajectory of the robot that minimizes the energy consumption during the motion. The presence of the waypoint constraints makes this optimal control problem particularly difficult to solve. The mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the waypoints part of the state, then introducing binary variables to enforce the constraint of passing once through each waypoint, and finally applying a fifth-degree Gauss–Lobatto direct collocation method to tackle the dynamic constraints. High-degree interpolation polynomials allow the number of variables of the problem to be reduced for a given numerical precision. The resulting mixed-integer nonlinear programming problem has been solved using a nonlinear programming-based branch-and-bound algorithm specifically tailored to the problem. The results of the numerical experiments have shown the effectiveness of the approach.     

Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization

We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow–Schwartz regularization method, which has already been applied to Markowitz portfolio problems.     

Polyhedral approximation in mixed-integer convex optimization

Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro.     

Lift-and-project cuts for convex mixed integer nonlinear programs

We describe a computationally effective method for generating lift-and-project cuts for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs and in the limit generates an inequality as strong as the lift-and-project cut that can be obtained from solving a cut-generating nonlinear program. Using this procedure, we are able to approximately optimize over the rank one lift-and-project closure for a variety of convex MINLP instances. The results indicate that lift-and-project cuts have the potential to close a significant portion of the integrality gap for convex MINLPs. In addition, we find that using this procedure within a branch-and-cut solver for convex MINLPs significantly reduces the total solution time for many instances. We also demonstrate that combining lift-and-project cuts with an extended formulation that exploits separability of convex functions yields significant improvements in both relaxation bounds and the time to calculate the relaxation. Overall, these results suggest that with an effective separation routine, like the one proposed here, lift-and-project cuts may be as effective for solving convex MINLPs as they have been for solving mixed-integer linear programs.     

Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems

Numerous planning problems can be formulated as multi-stage stochastic programs and many possess key discrete (integer) decision variables in one or more of the stages. Progressive hedging (PH) is a scenario-based decomposition technique that can be leveraged to solve such problems. Originally devised for problems possessing only continuous variables, PH has been successfully applied as a heuristic to solve multi-stage stochastic programs with integer variables. However, a variety of critical issues arise in practice when implementing PH for the discrete case, especially in the context of very difficult or large-scale mixed-integer problems. Failure to address these issues properly results in either non-convergence of the heuristic or unacceptably long run-times. We investigate these issues and describe algorithmic innovations in the context of a broad class of scenario-based resource allocation problem in which decision variables represent resources available at a cost and constraints enforce the need for sufficient combinations of resources. The necessity and efficacy of our techniques is empirically assessed on a two-stage stochastic network flow problem with integer variables in both stages.