Convex Mixed-integer Programming Research Articles

Lane’s Algorithm Revisited

In 1964, Kenneth Lane proposed an algorithm to optimize the production schedule of a single-metal, single-processor open pit mine. For this, he proposed a policy based on varying, over time, the so-called “cutoff grade”—or grade threshold used to determine if extracted material should be ore (processed material) or waste (thrown away). Lane’s algorithm had a profound impact on the mining industry. However, though it has been used in multiple commercial software systems and has traditionally been taught to every aspiring mining engineer, it is widely considered a heuristic, and little is known regarding the quality of the solutions it produces. In this paper, we formally study Lane’s problem. We show that Lane’s algorithm can be viewed as an approximate dynamic programming scheme and that Lane’s optimality conditions can be formally derived in two different ways: by considering a variant of the problem where the future value function is linearly approximated or by deriving the optimality conditions of a continuous-time version of the problem. We further show that Lane’s algorithm can naturally be extended to this continuous-time version of the problem and that when this algorithm converges, it converges to an optimal solution. Finally, through a reformulation, we show that Lane’s original problem can be solved using convex mixed-integer programming. Though hypothetical counterexamples can be constructed, computational experiments prove that Lane’s algorithm can produce the optimal solution in every real-world data set tested, thereby lending solid support for its practical application. This paper was accepted by Chung Piaw Teo, optimization.

Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares

Valid linear inequalities are substantial in linear and convex mixed-integer programming. This article deals with the computation of valid linear inequalities for nonlinear programs. Given a point in the feasible set, we consider the task of computing a tight valid inequality. We reformulate this geometrically as the problem of finding a hyperplane which minimizes the distance to the given point. A characterization of the existence of optimal solutions is given. If the constraints are given by polynomial functions, we show that it is possible to approximate the minimal distance by solving a hierarchy of sum of squares programs. Furthermore, using a result from real algebraic geometry, we show that the hierarchy converges if the relaxed feasible set is bounded. We have implemented our approach, showing that our ideas work in practice.

Feasible partition problem in reverse convex and convex mixed-integer programming

In this paper we consider the consistent partition problem in reverse convex and convex mixed-integer programming. In particular we will show that for the considered classes of convex functions, both integer and relaxed systems can be partitioned into two disjoint subsystems, each of which is consistent and defines an unbounded region. The polynomial time algorithm to generate the partition will be proposed and the algorithm for a maximal partition will also be provided.

Joint Scalable Coding and Routing for 60 GHz Real-Time Live HD Video Streaming Applications

Transmission of high-definition (HD) video is a promising application for 60 GHz wireless links, since very high transmission rates (up to several Gbit/s) are possible. In particular we consider a sports stadium broadcasting system where signals from multiple cameras are transmitted to a central location. Due to the high pathloss of 60 GHz radiation over the large distances encountered in this scenario, the use of relays might be required. The current paper analyzes the joint selection of the routes (relays) and the compression rates from the various sources for maximization of the overall video quality. We consider three different scenarios: (i) each source transmits only to one relay and the relay can receive only one data stream, and (ii) each source can transmit only to a single relay, but relays can aggregate streams from different sources and forward to the destination, and (iii) the source can split its data stream into parallel streams, which can be transmitted via different relays to the destination. For each scenario, we derive the mathematical formulations of the optimization problem and re-formulate them as convex mixed-integer programming, which can guarantee optimal solutions. Extensive simulations demonstrate that high-quality transmission is possible for at least ten cameras over distances of 300 m. Furthermore, optimization of the video quality gives results that can significantly outperform algorithms that maximize data rates.

The Retail Planning Problem Under Demand Uncertainty

We consider the Retail Planning Problem in which the retailer chooses suppliers, and determines the production, distribution and inventory planning for products with uncertain demand in order to minimize total expected costs. This problem is often faced by large retail chains that carry private label products. We formulate this problem as a convex mixed integer program and show that it is strongly NP-hard. We determine a lower bound by applying a Lagrangean relaxation and show that this bound outperforms the standard convex programming relaxation, while being computationally efficient. We also establish a worst-case error bound for the Lagrangean relaxation. We then develop heuristics to generate feasible solutions. Our computational results indicate that our convex programming heuristic yields feasible solutions that are close to optimal with an average suboptimality gap at 3.4%. We also develop managerial insights for practitioners who choose suppliers, and make production, distribution and inventory decisions in the supply chain.

Designing a stochastic distribution network model under risk

In this paper, we present a stochastic supply chain network model under risk, in which three tiers of decision makers of suppliers, distribution centers (DCs), and customers seek to determine their optimal plans. Unlike other studies in the extant literature, we use financial risk measure to model and for the first time, a new methodology based on finance literature is used to optimize the conditional value-at-risk (CVaR) measure for the problem. In fact the aim of this paper is to show the benefits of considering CVaR to control the risks in the supply chain networks. Also in order to obtain the economics of scale, a piecewise linear transportation cost function is applied between suppliers and DCs. We formulate the problem as a convex mixed integer program. A two-phase (constructive and improvement) heuristic method is developed to solve the problem. In the constructive phase, an initial solution is built randomly, and in improvement phase, this solution is improved iteratively by using a hybrid algorithm combining Tabu search and simulated annealing methods. Numerical experiments demonstrate the practicability of considering CVaR in the model which is promising in the supply chain network area, and also the significant efficiency of the heuristic method in terms of CPU times.

Fleet Management for Vehicle Sharing Operations

Astochastic, mixed-integer program (MIP) involving joint chance constraints is developed that generates least-cost vehicle redistribution plans for shared-vehicle systems such that a proportion of all near-term demand scenarios are met. The model aims to correct short-term demand asymmetry in shared-vehicle systems, where flow from one station to another is seldom equal to the flow in the opposing direction. The model accounts for demand stochasticity and generates partial redistribution plans in circumstances when demand outstrips supply. This stochastic MIP has a nonconvex feasible region. A novel divide-and-conquer algorithm for generating p-efficient points, used to transform the problem into a set of disjunctive, convex MIPs and handle dual-bounded chance constraints, is proposed. Assuming independence of random demand across stations, a faster cone-generation method is also presented. In a real-world application for a system in Singapore, the potential of redistribution as a fleet management strategy and the value of accounting for inherent stochasticities are demonstrated.

Feasibility in reverse convex mixed-integer programming

In this paper we address the problem of the infeasibility of systems defined by reverse convex inequality constraints, where some or all of the variables are integer. In particular, we provide a polynomial algorithm that identifies a set of all constraints critical to feasibility ( CF ) , that is constraints that may affect a feasibility status of the system after some perturbation of the right-hand sides. Furthermore, we will investigate properties of the irreducible infeasible sets and infeasibility sets, showing in particular that every irreducible infeasible set as well as infeasibility sets in the considered system, are subsets of the set CF of constraints critical to feasibility.

Global optimization for signomial discrete programming problems in engineering design

Signomial discrete programming (SDP) problems arise frequently in a variety of real applications. Although many optimization techniques have been developed to solve an SDP problem, they use too many binary variables to reformulate the problem for finding a globally optimal solution or can only derive a local or an approximate solution. This article proposes a global optimization method to solve an SDP problem by integrating an efficient linear expression of single variable discrete functions and convexification techniques. An SDP problem can be converted into a convex mixed-integer programming problem solvable to obtain a global optimum. Several illustrative examples are also presented to demonstrate the usefulness and effectiveness of the proposed method.

Finding all solutions of systems of nonlinear equations with free variables

Systems of nonlinear equations often represent mathematical models in engineering design. This study proposes a novel method for finding all solutions of systems of nonlinear equations with free variables. The original problem is first transformed into a global optimization problem whose multiple global minima with a zero objective value correspond to all solutions of the original problem. Then, by using variable substitution on free variables and applying convexification strategies and piecewise linearization techniques on nonconvex functions, the transformed optimization problem is reformulated as a convex mixed-integer program solvable to reach an approximately global optimum. An algorithm is developed to find all solutions of the reformulated problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method in engineering design.

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.