Two linear approximation algorithms for convex mixed integer nonlinear programming

We present two new algorithms for convex Mixed Integer Nonlinear Programming (MINLP), both based on the well known Extended Cutting Plane (ECP) algorithm proposed by Weterlund and Petersson. Our first algorithm, Refined Extended Cutting Plane (RECP), incorporates additional cuts to the MILP relaxation of the original problem, obtained by solving linear relaxations of NLP problems considered in the Outer Approximation algorithm. Our second algorithm, Linear Programming based Branch-and-Bound (LP-BB), applies the strategy of generating cuts that is used in RECP, to the linear approximation scheme used by the LP/NLP based Branch-and-Bound algorithm. Our computational results show that RECP and LP-BB are highly competitive with the most popular MINLP algorithms from the literature, while keeping the nice and desirable characteristic of ECP, of being a first-order method.

     

Parametric methods in integer linear programming

In contrast to methods of parametric linear programming which were developed soon after the invention of the simplex algorithm and are easily included as an extension of that method, techniques for parametric analysis on integer programs are not well known and require considerable effort to append them to an integer programming solution algorithm.The paper reviews some of the theory employed in parametric integer programming, then discusses algorithmic work in this area over the last 15 years when integer programs are solved by different methods. A summary of applications is included and the article concludes that parametric integer programming is a valuable tool of analysis awaiting further popularization.     

Marginal values in mixed integer linear programming

For a given optimization problem, P, considered as a function of the data, its marginal values are defined as the directional partial derivatives of the value of P with respect to perturbations in that data. For linear programs, formulas for the marginal values were given by Mills, [10], and further developed by the current author [16]. In this paper, the marginal value formulas are extended to the case of mixed integer linear programming (MIP). As in ordinary linear programming, discontinuities in the value can occur, and the analysis here identifies them. This latter aspect extends previous work on continuity by the current author, [18], Geoffrion and Nauss, [5], Nauss, [11], and Radke, [12], and work on the value function of Blair and Jeroslow, [2]. Application is made to model formulation and to post-optimal analysis.Supported in part by the Air Force Office of Scientific Research, Grant # AFSOR-0271 to Rutgers University.     

Integer convex minimization by mixed integer linear optimization

Minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension (Grötschel et al., 1988). We provide an alternative, short, and geometrically motivated proof of this result. In particular, we present an oracle-polynomial algorithm based on a mixed integer linear optimization oracle.     

Exact augmented Lagrangian duality for mixed integer linear programming

We investigate the augmented Lagrangian dual (ALD) for mixed integer linear programming (MIP) problems. ALD modifies the classical Lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap. We first provide a primal characterization for ALD for MIPs and prove that ALD is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity. This provides an alternative characterization and proof of a recent result in Boland and Eberhard (Math Program 150(2):491–509, 2015, Proposition 3). We further show that, under some mild conditions, ALD using any norm as the augmenting function is able to close the duality gap of an MIP with a finite penalty coefficient. This generalizes the result in Boland and Eberhard (2015, Corollary 1) from pure integer programming problems with bounded feasible region to general MIPs. We also present an example where ALD with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient.     

The use of linear programming duality in mixed integer programming

This paper is about the primal-dual relationship in a mixedinteger programming problem (MIP) in which integer variablesare binary. It shows how the primal-dual relationship of a linearprogramming problem (LP) can be used to advantage in MIPs. Thecentral idea is to look conceptually at the nature of all possibleLPs that arise from all possible settings for the discrete variablesin order to deduce general properties of the solution set. Afterdeveloping the relevant theory, we show the usefulness of thisaproach by applying it to three totally different problems.New results are derived for the method of least median of squaresin robust regression, the problem of rectilinear obnoxious-facilitylocation, and the problem of finding a fixed-size rectanglecontaining the minimum weight of points.     

Scheduling staff using mixed integer programming

This paper describes the solution of a problem of scheduling a workforce so as to meet demand which varies markedly with the time of day and moderately with the day of week. The main objectives are determining how many staff to employ and the times at which shifts should start. The problem is expressed as a large MIP problem initially presenting computational difficulties. The difficulties vanish when the formulation is modified and a package allowing the use of reduce and (especially) special ordered sets becomes available. The client has commissioned the study primarily to benchmark its existing schedule by comparing it with a theoretical optimum. The optimal schedule and comparison are very sensitive to technical and cost coefficients which are not precisely known.     

Heuristic algorithms for the inverse mixed integer linear programming problem

The recent development in inverse optimization, in particular the extension from the inverse linear programming problem to the inverse mixed integer linear programming problem (InvMILP), provides more powerful modeling tools but also presents more challenges to the design of efficient solution techniques. The only reported InvMILP algorithm, referred to as Alg^InvMILP, although finitely converging to global optimality, suffers two limitations that greatly restrict its applicability: it is time consuming and does not generate a feasible solution except for the optimal one. This paper presents heuristic algorithms that are designed to be implemented and executed in parallel with Alg^InvMILP in order to alleviate and overcome its two limitations. Computational experiments show that implementing the heuristic algorithm on one auxiliary processor in parallel with Alg^InvMILP on the main processor significantly improves its computational efficiency, in addition to providing a series of improving feasible upper bound solutions. The additional speedup of parallel implementation on two or more auxiliary processors appears to be incremental, but the upper bound can be improved much faster.     

An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

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.     

Complete mixed integer linear programming formulations for modularity density based clustering

Modularity density maximization is a clustering method that improves some issues of the commonly used modularity maximization approach. Recently, some Mixed-Integer Linear Programming (MILP) reformulations have been proposed in the literature for the modularity density maximization problem, but they require as input the solution of a set of auxiliary binary Non-Linear Programs (NLPs). These can become computationally challenging when the size of the instances grows. In this paper we propose and compare some explicit MILP reformulations of these auxiliary binary NLPs, so that the modularity density maximization problem can be completely expressed as MILP. The resolution time is reduced by a factor up to two order of magnitude with respect to the one obtained with the binary NLPs.     

A multi-objective mixed integer linear programming model for thesis defence scheduling

In this paper, we address the thesis defence scheduling problem, a critical academic scheduling management process, which has been overshadowed in the literature by its counterparts, course timetabling and exam scheduling. Specifically, we address the single defence assignment type of thesis defence scheduling problems, where each committee is assigned to a single defence, scheduled for a specific day, hour and room. We formulate a multi-objective mixed-integer linear programming model, which aims to be applicable to a broader set of cases than other single defence assignment models present in the literature, which have a focus on the characteristics of their universities. For such a purpose, we introduce a different decision variable, propose constraint formulations that are not regulation and policy specific, and cover and offer new takes on the more common objectives seen in the literature. We also include new objective functions based on our experience with the problem at our university and by applying knowledge from other academic scheduling problems. We also propose a two-stage solution approach. The first stage is employed to find the number of schedulable defences, enabling the optimisation of instances with unschedulable defences. The second stage is an implementation of the augmented $ϵ$-constraint method, which allows for the search of a set of different and non-dominated solutions while skipping redundant iterations. The methodology is tested for case-studies from our university, significantly outperforming the solutions found by human schedulers. A novel instance generator for thesis scheduling problems is presented. Its main benefit is the generation of the availability of committee members and rooms in availability and unavailability blocks, resembling their real-world counterparts. A set of 96 randomly generated instances of varying sizes is solved and analysed regarding their relative computational performance, the number of schedulable defences and the distribution of the considered types of iterations. The proposed method can find the optimal number of schedulable defences and present non-dominated solutions within the set time limits for every tested instance.     

A mixed integer linear programming model for optimal sovereign debt issuance

Governments borrow funds to finance the excess of cash payments or interest payments over receipts, usually by issuing fixed income debt and index-linked debt. The goal of this work is to propose a stochastic optimization-based approach to determine the composition of the portfolio issued over a series of government auctions for the fixed income debt, to minimize the cost of servicing debt while controlling risk and maintaining market liquidity. We show that this debt issuance problem can be modeled as a mixed integer linear programming problem with a receding horizon. The stochastic model for the interest rates is calibrated using a Kalman filter and the future interest rates are represented using a recombining trinomial lattice for the purpose of scenario-based optimization. The use of a latent factor interest rate model and a recombining lattice provides us with a realistic, yet very tractable scenario generator and allows us to do a multi-stage stochastic optimization involving integer variables on an ordinary desktop in a matter of seconds. This, in turn, facilitates frequent re-calibration of the interest rate model and re-optimization of the issuance throughout the budgetary year allows us to respond to the changes in the interest rate environment. We successfully demonstrate the utility of our approach by out-of-sample back-testing on the UK debt issuance data.     

Cutting plane algorithms for the inverse mixed integer linear programming problem

We present cutting plane algorithms for the inverse mixed integer linear programming problem (InvMILP), which is to minimally perturb the objective function of a mixed integer linear program in order to make a given feasible solution optimal.     

A computational study of an objective hyperplane search heuristic for the general integer linear programming problem

The paper describes an objective function hyperplane search heuristic for solving the general all-integer linear programming problem (ILP). The algorithm searches a series of objective function hyperplanes and the search over any given hyperplane is formulated as a bounded knapsack problem. Theory developed for combinations of the objective function and problem constraints is used to guide the search. We evaluate the algorithm's performance on a class of ILP problems to assess the areas of effectiveness.     

The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.     

Cross decomposition for mixed integer programming

Many methods for solving mixed integer programming problems are based either on primal or on dual decomposition, which yield, respectively, a Benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via Lagrangean relaxation. These methods exploit either the primal or the dual structure of the problem. We propose a new approach, cross decomposition, which allows exploiting simultaneously both structures. The development of the cross decomposition method captures profound relationships between primal and dual decomposition. It is shown that the more constraints can be included in the Langrangean relaxation (provided the duality gap remains zero), the fewer the Benders cuts one may expect to need. If the linear programming relaxation has no duality gap, only one Benders cut is needed to verify optimality.     

Heuristics for convex mixed integer nonlinear programs

In this paper, we describe the implementation of some heuristics for convex mixed integer nonlinear programs. The work focuses on three families of heuristics that have been successfully used for mixed integer linear programs: diving heuristics, the Feasibility Pump, and Relaxation Induced Neighborhood Search (RINS). We show how these heuristics can be adapted in the context of mixed integer nonlinear programming. We present results from computational experiments on a set of instances that show how the heuristics implemented help finding feasible solutions faster than the traditional branch-and-bound algorithm and how they help in reducing the total solution time of the branch-and-bound algorithm.     

Models and algorithms to improve earthwork operations in road design using mixed integer linear programming

In road construction, earthwork operations account for about 25% of the construction costs. Existing linear programming models for earthwork optimization are designed to minimize the hauling costs and to balance the earth across the construction site. However, these models do not consider the removal of physical blocks that may influence the earthwork process. As such, current models may result in inaccurate estimates of optimal earthwork costs, leading to poor choices in road design. In this research, we extend the classical linear program model of earthwork operations to a mixed integer linear program model that accounts for blocks. We examine the economic impact of incorporating blocks via mixed integer linear programming, and find significant savings for most road designs in our test-set. However, the resulting model is considerably harder to solve than the original linear program. Based on structural observations, we introduce a set of algorithms that theoretically reduce the solving time of the model. We confirm this reduction in solve time with numerical experiments.     

On the global solution of multi-parametric mixed integer linear programming problems

This paper deals with the global solution of the general multi-parametric mixed integer linear programming problem with uncertainty in the entries of the constraint matrix, the right-hand side vector, and in the coefficients of the objective function. To derive the piecewise affine globally optimal solution, the steps of a multi-parametric branch-and-bound procedure are outlined, where McCormick-type relaxations of bilinear terms are employed to construct suitable multi-parametric under- and overestimating problems. The alternative of embedding novel piecewise affine relaxations of bilinear terms in the proposed algorithmic procedure is also discussed.