In this paper, we consider the canonical water network design problem, which contains nonconvex potential loss functions and discrete resistance choices with varying costs. Traditionally, to resolve the nonconvexities of this problem, relaxations of the potential loss constraints have been applied to yield a more tractable mixed-integer convex program (MICP). However, design solutions to these relaxed problems may not be feasible with respect to the full nonconvex physics. In this paper, it is shown that, in fact, the original mixed-integer nonconvex program can be reformulated exactly as an MICP. Beginning with a convex program previously used for proving nonlinear network design feasibility, strong duality is invoked to construct a novel, convex primal-dual system embedding all physical constraints. This convex system is then augmented to form an exact MICP formulation of the original design problem. Using this novel MICP as a foundation, a global optimization algorithm is developed, leveraging heuristics, outer approximations, and feasibility cutting planes for infeasible designs. Finally, the algorithm is compared against the previous relaxation-based state of the art in water network design over a number of standard benchmark instances from the literature.
We obtain the dynamic programming equations and optimality conditions akin to Pontryagin's extremum principle for certain mathematical models of hybrid control systems.
Quantum computing is emerging as a new computing resource that could be superior to conventional computing for certain classes of optimization problems. However, in principle, most existing approaches to quantum optimization are intended to solve unconstrained binary programming problems, while mixed-integer linear programming is of most interest in practice. We attempt to bridge the gap between the capability of quantum computing and real-world applications by developing a new approach for mixed-integer programming. The approach applies Benders decomposition to decompose the mixed-integer pro…
This paper presents a mixed-integer quadratic programming formulation of an existing data-driven approach to computational elasticity. This formulation is suitable for application of a standard mixed-integer programming solver, which finds a global optimal solution. Therefore, the results obtained by the presented method can be used as benchmark instances for any other algorithm. Preliminary numerical experiments are performed to compare quality of solutions obtained by the proposed method and a heuristic used in the data-driven computational mechanics.
In this work, we present a modeling framework for minimax mixed 0-1 fuzzy linear problems. It is based on extending the usual rewriting of crisp minimax problems via auxiliary variables to model the maximum of a finite set of fuzzy linear functions. We establish that the considered problem can be equivalently formulated as a multiple objective mixed integer programming problem. The framework is applied to a fully fuzzy version of the capacitated center facility location problem.
Multi-objective integer or mixed-integer programming problems typically have disconnected feasible domains, making the task of constructing an approximation of the Pareto front challenging. The present paper shows that certain algorithms which were originally devised for continuous problems can be successfully adapted to approximate the Pareto front for integer, and mixed-integer, multi-objective optimization problems. Relationships amongst various scalarization techniques are established to motivate the choice of a particular scalarization in these algorithms. The proposed algorithms are test…
We consider the constrained assortment optimization problem under the mixed multinomial logit model. Even moderately sized instances of this problem are challenging to solve directly using standard mixed-integer linear optimization formulations. This has motivated recent research exploring customized optimization strategies and approximation techniques. In contrast, we develop a novel conic quadratic mixed-integer formulation. This new formulation, together with McCormick inequalities exploiting the capacity constraints, enables the solution of large instances using commercial optimization sof…
We present a method for solving the general mixed constrained convex quadratic programming problem using an active set method on the dual problem. The approach is similar to existing active set methods, but we present a new way of solving the linear systems arising in the algorithm. There are two main contributions; we present a new way of factorizing the linear systems, and show how iterative refinement can be used to achieve good accuracy and to solve both types of sub-problems that arise from semi-definite problems.
A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objecti…
This paper addresses the single-item single-stocking location stochastic lot sizing problem under the $(s, S) $ policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal $(s, S)$ policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimisation software. Computational experiments demonstrate that optimality gaps of t…
We give an explicit geometric way to build mixed-integer programming (MIP) formulations for unions of polyhedra. The construction is simply described in terms of spanning hyperplanes in an r-dimensional linear space. The resulting MIP formulation is ideal, and uses exactly r integer variables and 2 x (# of spanning hyperplanes) general inequality constraints. We use this result to derive novel logarithmic-sized ideal MIP formulations for discontinuous piecewise linear functions and structures appearing in robotics and power systems problems.
Mixed integer quadratic programming (MIQP) is the problem of minimizing a convex quadratic function over mixed integer points in a rational polyhedron. This paper focuses on the augmented Lagrangian dual (ALD) for MIQP. ALD augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function…
It is a crucial problem how to heat oil and save running cost for crude oil transport. This paper strictly formulates such a heated oil pipeline problem as a mixed integer nonlinear programming model. Nonconvex and convex continuous relaxations of the model are proposed, which are proved to be equivalent under some suitable conditions. Meanwhile, we provide a preprocessing procedure to guarantee these conditions. Therefore we are able to design a branch-and-bound algorithm for solving the mixed integer nonlinear programming model to global optimality. To make the branch-and-bound algorithm mor…
We propose a feasible active set method for convex quadratic programming problems with non-negativity constraints. This method is specifically designed to be embedded into a branch-and-bound algorithm for convex quadratic mixed integer programming problems. The branch-and-bound algorithm generalizes the approach for unconstrained convex quadratic integer programming proposed by Buchheim, Caprara and Lodi to the presence of linear constraints. The main feature of the latter approach consists in a sophisticated preprocessing phase, leading to a fast enumeration of the branch-and-bound nodes. Mor…
In this work, the online printing shop scheduling problem is considered. This challenging real problem, that appears in the nowadays printing industry, can be seen as a flexible job shop scheduling problem with sequence flexibility in which precedence constraints among operations of a job are given by an arbitrary directed acyclic graph. In addition, several complicating particularities such as periods of unavailability of the machines, resumable operations, sequence-dependent setup times, partial overlapping among operations with precedence constraints, release times, and fixed operations are…
In this paper, we establish a new approach to solve the tensor complementarity problem (TCP). A mixed integer programming model is given and the TCP is solved by solving the model. The TCP is shown to be formulated as an equivalent mixed integer feasibility problem. Based on the reformulation, some conditions are obtained to guarantee the solvability of the TCP. Specially, a sufficient condition is given for TCP without solutions. A necessary and sufficient condition is given for existence of solutions. We also give a concrete bound for the solution set of the TCP with positive definite tensor…
Shunt FACTS devices, such as, a Static Var Compensator (SVC), are capable of providing local reactive power compensation. They are widely used in the network to reduce the real power loss and improve the voltage profile. This paper proposes a planning model based on mixed integer conic programming (MICP) to optimally allocate SVCs in the transmission network considering load uncertainty. The load uncertainties are represented by a number of scenarios. Reformulation and linearization techniques are utilized to transform the original non-convex model into a convex second order cone programming (…
We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representa…
In this paper, we mainly study one class of convex mixed-integer nonlinear programming problems (MINLPs) with non-differentiable data. By dropping the differentiability assumption, we substitute gradients with subgradients obtained from KKT conditions, and use the outer approximation method to reformulate convex MINLP as one equivalent MILP master program. By solving a finite sequence of subproblems and relaxed MILP problems, we establish an outer approximation algorithm to find the optimal solution of this convex MINLP. The convergence of this algorithm is also presented. The work of this pap…
In this paper, a mixed integer linear programming (MILP) formulation is proposed to solve the dynamic economic dispatch with valve-point effect (DED-VPE). Based on piecewise linearization technique, the non-convex and non-smooth generation cost is reformulated into a linear lower approximation which is better than the quadratic one, yielding an MILP formulation for the DED-VPE. When the segment parameter is set appropriately, the MILP formulation can be solved by a mixed integer programming (MIP) solver directly and efficiently. Thus, a global optimal solution within a preset tolerance can be …
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…
This is Part II of a study on mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We set the focus on MIP relaxation methods for non-convex continuous variable products and extend the well-known MIP relaxation normalized multi-parametric disaggregation technique (NMDT), applying a sophisticated discretization to both variables. We refer to this approach as doubly discretized normalized multiparametric disaggregation technique (D-NMDT). In a comprehensive theoretical analysis, we underline the…
For central banks, managing the minting is one of the most important task since a shortage yields negative economic and social impacts, and the budget committed for minting is one of the largest within the central banks. Hence, the central bank requires to find the mixture of coins to be produced that satisfies the demand, inventory and production constraints while minimizing the cost. We propose a mixed-integer programming model that minimize the cost of minting by reducing the number of extra-shifts required while fulfilling the constraints. We also perform a simulation with data of a centra…
The traditional demand-responsive paratransit system plays an important role in connecting people to health care, particularly those who are carless, low-income, senior, underinsured/uninsured, or who have a disability. However, the existing paratransit system usually has low service quality due to long waiting times, low operation frequencies, and high costs. In order to improve the service quality, we propose to design a new demand-responsive paratransit system that offers public, Uber-like options for non-emergency medical transportation. Mixed integer programming models are thus developed …
Mixed-integer linear programming (MILP) is widely employed for modeling combinatorial optimization problems. In practice, similar MILP instances with only coefficient variations are routinely solved, and machine learning (ML) algorithms are capable of capturing common patterns across these MILP instances. In this work, we combine ML with optimization and propose a novel predict-and-search framework for efficiently identifying high-quality feasible solutions. Specifically, we first utilize graph neural networks to predict the marginal probability of each variable, and then search for the best f…
This paper studies the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing the true distribution with a high statistical guarantee. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumption…
The Benders' decomposition algorithm is a technique in mathematical programming for complex mixed-integer linear programming (MILP) problems with a particular block structure. The strategy of Benders' decomposition can be described as a strategy of divide and conquer. The Benders' decomposition algorithm has been employed in a variety of applications such as communication, networking, and machine learning. However, the master problem in Benders' decomposition is still NP-hard, which motivates us to employ quantum computing. In the paper, we propose a hybrid quantum Benders' decomposition algor…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are amenable to exact copositive programming reformulations of polynomial size. These convex optimization problems are NP-hard but admit a conservative semidefinite programming (SDP) approximation that can be solved efficiently. We prove that the popular approximate S-lemma method --- which is valid only in the case of continuous uncertainty --- is weaker than our a…
A key ingredient in branch and bound (B&B) solvers for mixed-integer programming (MIP) is the selection of branching variables since poor or arbitrary selection can affect the size of the resulting search trees by orders of magnitude. A recent article by Le Bodic and Nemhauser [Mathematical Programming, (2017)] investigated variable selection rules by developing a theoretical model of B&B trees from which they developed some new, effective scoring functions for MIP solvers. In their work, Le Bodic and Nemhauser left several open theoretical problems, solutions to which could guide the future d…
We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks. These formulations can be used for a number of important tasks, such as verifying that an image classification network is robust to adversarial inputs, or solving decision problems where the objective function is a machine learning model. We present a generic framework, which may be of independent interest, that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domai…
We address the problem of scheduling water resources in a power system via approximate dynamic programming.To this goal, we model a finite horizon economic dispatch problemwith convex stage cost and affine dynamics, and consider aquadratic approximation of the value functions. Evaluating theachieved policy entails solving a quadratic program at each timestep, while value function fitting can be cast as a semidefiniteprogram. We test our proposed algorithm on a simplified versionof the Uruguayan power system, achieving a four percent costreduction with respect to the myopic policy
This paper investigates the water network's potential ability to provide demand response services to the power grid under the framework of a distribution-level water-energy nexus (micro-WEN). In particular, the hidden controllability of water loads, such as irrigation systems, was closely studied to improve the flexibility of electrical grids. A optimization model is developed for the demand-side management (DSM) of micro-WEN, and the simulation results assert that grid flexibility indeed benefits from controllable water loads. Although the proposed optimal DSM model is an intractable mixed-in…
In this paper we propose a fast optimization algorithm for approximately minimizing convex quadratic functions over the intersection of affine and separable constraints (i.e., the Cartesian product of possibly nonconvex real sets). This problem class contains many NP-hard problems such as mixed-integer quadratic programming. Our heuristic is based on a variation of the alternating direction method of multipliers (ADMM), an algorithm for solving convex optimization problems. We discuss the favorable computational aspects of our algorithm, which allow it to run quickly even on very modest comput…
Mixed integer nonlinear programs (MINLPs) are arguably among the hardest optimization problems, with a wide range of applications. MINLP solvers that are based on linear relaxations and spatial branching work similar as mixed integer programming (MIP) solvers in the sense that they are based on a branch-and-cut algorithm, enhanced by various heuristics, domain propagation, and presolving techniques. However, the analysis of infeasible subproblems, which is an important component of most major MIP solvers, has been hardly studied in the context of MINLPs. There are two main approaches for infea…
We propose an early termination technique for mixed integer conic programming for use within branch-and-bound based solvers. Our approach generalizes previous early termination results for ADMM-based solvers to a broader class of primal-dual algorithms, including both operator splitting methods and interior point methods. The complexity for checking early termination is $O(n)$ for each termination check assuming a bounded problem domain. We show that this domain restriction can be relaxed for problems whose data satisfies a simple rank condition, in which case each check requires an $O(n^2)$ s…
Two-stage stochastic mixed-integer programming (SMIP) problems with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing the subset of the convex hull of feasible integer points of the second-stage subproblem which can be used for solving the SMIP. The basic idea is to consider a small enough subset of feasible integer points that is necessary for generating a valid inequality for the integer subproblem. An algorithm for obtaining such a subset based on the solution of the subproblem LP-relaxa…
We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new algorithm is an inner loop of optimized linearization steps, similar to those taken in the classical F…
Low levels of system inertia in power grids with significant penetration of non-synchronous Renewable Energy Sources (RES) have increased the risk of frequency instability. The provision of a certain type of ancillary services such as inertia and Frequency Response (FR) is needed at all times, to maintain system frequency within secure limits if the loss of a large power infeed were to occur. In this paper we propose a frequency-secured optimisation framework for the procurement of inertia and FR with diverse dynamics, which enables to apply a marginal-pricing scheme for these services. This p…
We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based am…
Many problems of interest for cyber-physical network systems can be formulated as Mixed Integer Linear Programs in which the constraints are distributed among the agents. In this paper we propose a distributed algorithm to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. In the proposed algorithm, at each communication round, agents solve locally a small LP, generate suitable cutting planes, namely intersection cuts and cost-based cuts, and communicate a fixed number of active constraints, i.e.,…
We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky, formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in the approximation error. Combining this with a diagonal perturbation technique to convert a nonseparable quadratic function into a separable one, we present a mixe…
The analysis of infeasible subproblems plays an import role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems. The first is to analyze the sequence of implications obtained by domain propagation that led to infeasibility. The result of the analysis are one or more sets of contradicting variable bounds from which so-called conflict constraints can be generated. This concept has its origin in solving satisfiability problems and is similarly used in …
We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most useful for hybrid model predictive control, where on-line implementation is hampered by the worst-case exponential complexity of mixed-integer solvers. The output is a simplicial partition which defines a static map from the current state to a suboptimal solution. The primary theoretical contribution of this paper is to introduce a non-zero optimal cost over…
Two essential ingredients of modern mixed-integer programming (MIP) solvers are diving heuristics that simulate a partial depth-first search in a branch-and-bound search tree and conflict analysis of infeasible subproblems to learn valid constraints. So far, these techniques have mostly been studied independently: primal heuristics under the aspect of finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. Here, we combine both concepts in two different ways. First, we develop a diving h…
In this paper, we present the first outer approximation algorithm for multi-objective mixed-integer linear programming problems with any number of objectives. The algorithm also works for certain classes of non-linear programming problems. It produces the non-dominated extreme points as well as the facets of the convex hull of these points. The algorithm relies on an oracle which solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the non-dominated extreme points in the case of multiobjec…
