Reformulation-linearization Technique Research Articles

Convexification techniques for linear complementarity constraints

We develop convexification techniques for mathematical programs with complementarity constraints. Specifically, we adapt the reformulation-linearization technique of Sherali and Adams (SIAM J Discrete Math 3, 411–430, 1990) to problems with linear complementarity constraints and discuss how this procedure reduces to its standard specification for binary mixed-integer programs. Then, we consider specially structured complementarity sets that appear in KKT systems with linear constraints. For sets with a single complementarity constraint, we develop a convexification procedure that generates all nontrivial facet-defining inequalities and has an appealing “cancel-and-relax” interpretation. This procedure is used to describe the convex hull of problems with few side constraints in closed-form. As a consequence, we delineate cases when the factorable relaxation techniques yield the convex hull from those for which they do not. We then discuss how these results extend to sets with multiple complementarity constraints. In particular, we show that a suitable sequential application of the cancel-and-relax procedure produces all nontrivial inequalities of their convex hull. We conclude by illustrating, on a set of randomly generated problems, that the relaxations we propose can be significantly stronger than those available in the literature.

Beamforming Optimization for Intelligent Reflecting Surface-Aided SWIPT IoT Networks Relying on Discrete Phase Shifts

Intelligent reflecting surface (IRS) is capable of constructing the favorable wireless propagation environment by leveraging massive low-cost reconfigurable reflect array elements. In this article, we investigate the IRS-aided multiple-input-multiple-output (MIMO) simultaneous wireless information and power transfer (SWIPT) for Internet-of-Things (IoT) networks, where the active base station (BS) transmits beamforming and the passive IRS reflection coefficients are jointly optimized for maximizing the minimum signal-to-interference-plus-noise ratio (SINR) among all information decoders (IDs), while maintaining the minimum total harvested energy at all energy receivers (ERs). Moreover, the IRS with practical discrete phase shifts is considered, and thereby the max-min SINR problem becomes an NP-hard combinatorial optimization problem with a strong coupling among optimization variables. To explore the insights and generality of this max-min design, both the single-ID single-ER (SISE) scenario and the multiple-IDs multiple-ERs (MIME) scenario are studied. In the SISE scenario, the classical combinatorial optimization techniques, namely, the special ordered set of type 1 (SOS1) and the reformulation-linearization (RL) technique, are applied to overcome the difficulty of this max-min design imposed by discrete optimization variables. Then, the optimal branch-and-bound algorithm and suboptimal alternating optimization algorithm are, respectively, proposed. We further extend the idea of alternating optimization to the MIME scenario. Moreover, to reduce the iteration complexity, a two-stage scheme is considered aiming to separately optimize the BS transmit beamforming and the IRS reflection coefficients. Finally, numerical simulations demonstrate the superior performance of the proposed algorithms over the benchmarks in both the two scenarios.

Container slot allocation and dynamic pricing of time-sensitive cargoes considering port congestion and uncertain demand

This paper studies the container slot allocation problem for time-sensitive cargoes with consideration of dynamic pricing and port congestion under uncertain demand. Time-sensitive cargoes call for express delivery. However, port congestion is a non-negligible factor to affect the delivery time. Hence, a new freight rate function with respect to the delivery time considering port congestion is proposed for time-sensitive cargoes, and two different slot allocation strategies are proposed: Loyal Strategy and Expansive Strategy. Accordingly, we formulate the problem as a one-stage container slot allocation model and a two-stage container slot allocation model, respectively. Both of the two models are stochastic mixed-integer quadratic programming models with chance constraints due to the involvement of dynamic pricing, port congestion, and uncertain demand. To solve the proposed models, a tailored algorithm that combines the Sample Average Approximation approach and the Reformulation Linearization Technique (SAA-RLT) is developed in this paper. Finally, numerical experiments are carried out to verify the applicability and effectiveness of the proposed models and the solution algorithm.

Models and algorithms for the product pricing with single-minded customers requesting bundles

We analyze a product pricing problem with single-minded customers, each interested in buying a bundle of products, if the total price is less than or equal to their budget. The objective of the seller is to maximize the total revenue and we assume that supply is unlimited for all products. We contribute to a missing piece of literature by giving some mathematical formulations for this single-minded bundle pricing problem. We first present a mixed-integer nonlinear program with bilinear terms in the objective function and the constraints. By applying classical linearization techniques, we obtain two different mixed-integer linear reformulations. Inspired by a reformulation–linearization technique framework, we derive valid inequalities leading to a tighter linear reformulation. We then discuss a Benders decomposition to project strong cuts from the tightest model onto a light and fast model. In another attempt to exploit the tighter linear relaxation, we discuss heuristics based on the developed mixed-integer linear formulations to quickly find good solutions. We conclude this work with extensive numerical experiments to assess the quality of the mixed-integer linear formulations, as well as the performance of the Benders decomposition and the heuristics.

Integration of wind generation uncertainties into frequency dynamic constrained unit commitment considering reserve and plug in electric vehicles

Increased penetration of renewable resources in power systems and consequently cut back in the inertia of the whole power system is a significant concern for Unit Commitment (UC) in terms of energy scheduling and frequency dynamics management. The new modeling of the UC problem presented in this paper provides optimal scheduling of the energy and reserve considering the frequency dynamics of the power system. In this work, impact of Wind Turbines (WT) as the clean generation and Plug-in Electric Vehicles (PEV) as the energy storage system are investigated considering a less conservative probabilistic modeling uncertainty of the wind. The paper targets to protect the security of frequency dynamics taking into account the Demand Response (DR) program and contribution of PEVs succeeding a generation loss. The Frequency Dynamics-constrained Unit Commitment (FDUC) considering PEVs was formulated as a Mixed-Integer Non-Linear Programming (MINLP) problem taking into consideration DR and WTs uncertainties. The proposed MINLP problem was then reformulated by the Reformulation-Linearization Technique (RLT) to derive a Mixed-Integer Linear Programming (MILP) problem. An IEEE 6-bus power system was served as a test system to evaluate the proposed approach and simulation results of four different Case study scenarios were obtained. Results revealed that the proposed approach can ensure frequency security and reduce the operational costs.

A reformulation-linearization based algorithm for the smallest enclosing circle problem

<p style='text-indent:20px;'>In this paper, an effective algorithm based on the reformulation-linearization technique (RLT) is developed to solve the smallest enclosing circle problem. Extensive computational experiments demonstrate that the algorithm based on the RLT outperforms the existing algorithms in terms of the solution time and quality in average.</p>

Multi-period optimal energy flow for electricity-gas integrated systems considering gas inertia and wind power uncertainties

The increasing integration of GFUs promotes the coordinated operation of power and gas systems to enhance the system efficiency and reliability. However, the gas inertia characteristic could have imperative impacts on the coordinated scheduling and this property has not been fully studied. This paper proposes a novel multi-period optimal energy flow (OEF) model considering gas inertia and wind power uncertainties for the scheduling and operation of integrated systems. The nonlinear transient natural gas transmission model is reformulated into a linear programming/quadratic programming (LP/QP) model via the reformulation linearization technique (RLT), yielding computational efficiency improvement. The gas inertia is molded as a time delay to study its impacts on the coordinated scheduling. Then, information gap decision theory (IGDT) based risk averse strategy is advocated to deal with the wind power uncertainties and the impact of the uncertainty level is quantified. Extensive simulation results on the modified 6-bus power system with 6-node gas system and the IEEE 39-bus power system integrated with a realistic 20-node natural gas network demonstrate the effectiveness of the proposed method.

A Linear Programming Approximation of Distributionally Robust Chance-Constrained Dispatch With Wasserstein Distance

This paper proposes a data-driven distributionally robust chance constrained real-time dispatch (DRCC-RTD) considering renewable generation forecasting errors. The proposed DRCC-RTD model minimizes the expected quadratic cost function and guarantees that the two-sided chance constraints are satisfied for any distribution in the ambiguity set. The Wasserstein-distance-based ambiguity set, which is a family of distributions centered at an empirical distribution, is employed to hedge against data perturbations. By applying the reformulation linearization technique (RLT) to relax the quadratic constraints of the worst-case costs and constructing linear reformulations of the DRCCs, the proposed DRCC-RTD model is cast into a deterministic linear programming (LP) problem, which can be solved efficiently by off-the-shelf solvers. Case studies are carried out on a 6-bus system and the IEEE 118-bus system to validate the effectiveness and efficiency of the proposed approach.

Source localization using TDOA and FDOA measurements based on semidefinite programming and reformulation linearization

The problem of source localization using time-difference-of-arrival (TDOA) and frequency-difference-of-arrival (FDOA) measurements has been widely studied. It is commonly formulated as a weighted least squares (WLS) problem with quadratic equality constraints. Due to the nonconvex nature of this formulation, it is difficult to produce a global solution. To tackle this issue, semidefinite programming (SDP) is utilized to convert the WLS problem to a convex optimization problem. However, the SDP-based methods will suffer obvious performance degradation when the noise level is high. In this paper, we devise a new localization solution using the SDP together with reformulation-linearization technique (RLT). Specifically, we firstly apply the RLT strategy to convert the WLS problem to a convex problem, and then add the SDP constraint to tighten the feasible region of the resultant formulation. Moreover, this solution is also extended for cases when there are sensor position and velocity errors. Numerical results show that our solution has significant accuracy advantages over the existing localization schemes at high noise levels.

Level 2 Reformulation Linearization Technique–Based Parallel Algorithms for Solving Large Quadratic Assignment Problems on Graphics Processing Unit Clusters

This paper discusses efficient parallel algorithms for obtaining strong lower bounds and exact solutions for large instances of the quadratic assignment problem (QAP). Our parallel architecture is comprised of both multicore processors and compute unified device architecture–enabled NVIDIA graphics processing units (GPUs) on the Blue Waters Supercomputing Facility at the University of Illinois at Urbana–Champaign. We propose novel parallelization of the Lagrangian dual ascent algorithm on the GPUs, which is used for solving a QAP formulation based on the level-2 reformulation linearization technique. The linear assignment subproblems in this procedure are solved using our accelerated Hungarian algorithm [Date K, Rakesh N (2016) GPU-accelerated Hungarian algorithms for the linear assignment problem. Parallel Computing 57:52–72.]. We embed this accelerated dual-ascent algorithm in a parallel branch-and-bound scheme and conduct extensive computational experiments on single and multiple GPUs, using problem instances with up to 42 facilities from the quadratic assignment problem library (QAPLIB). The experiments suggest that our GPU-based approach is scalable, and it can be used to obtain tight lower bounds on large QAP instances. Our accelerated branch-and-bound scheme is able to comfortably solve Nugent and Taillard instances (up to 30 facilities) from the QAPLIB, using a modest number of GPUs.

Allocating nodes to hubs for minimizing the hubs processing resources: A case study

This paper addresses the problem of allocating the terminal nodes to the hub nodes in a telecommunication network. Since the flow processing induces some undesirable delay, the objective is to minimize the total flow processed by the hubs. This study focuses on a real life network of the tunisian operator Tunisie Telecom whose operations managers are concerned by the quality of service. We provide three compact formulations that give optimal solutions for networks of large size. In particular, the last two are obtained by applying the Reformulation-Linearization Technique to a nonlinear formulation of the problem. The latter formulation derived within this approach is the most computationally effective, as pointed out by the computational experiments conducted on the real life network of Tunisie Telecom with 110 nodes and 5 hubs. Finally, we discuss and compare between the single allocation and double allocation configurations.

Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming

In this paper, we present new convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP) problems. While recent research has focused on strengthening convex relaxations of QCQP using the reformulation-linearization technique (RLT), the state-of-the-art methods lose their effectiveness when dealing with (multiple) nonconvex quadratic constraints in QCQP, except for direct lifting and linearization. In this research, we decompose and relax each nonconvex constraint to two second order cone (SOC) constraints and then linearize the products of the SOC constraints and linear constraints to construct some new effective valid constraints. Moreover, we extend the reach of the RLT-like techniques for almost all different types of constraint-pairs (including valid inequalities by linearizing the product of a pair of SOC constraints, and the Hadamard product or the Kronecker product of two respective valid linear matrix inequalities), examine dominance relationships among different valid inequalities, and explore almost all possibilities of gaining benefits from generating valid constraints. We also successfully demonstrate that applying RLT-like techniques to additional redundant linear constraints could reduce the relaxation gap significantly. We demonstrate the efficiency of our results with numerical experiments.

Reformulation, linearization, and decomposition techniques for balanced distributed operating room scheduling

We study the balanced distributed operating room (OR) scheduling (BDORS) problem as a location-allocation model, encompassing two levels of balancing decisions: (i) daily macro imbalance among collaborating hospitals in terms of the number of allocated ORs and (ii) daily micro imbalance among open ORs in each hospital in terms of the total caseload assigned. BDORS is formulated as a novel mixed-integer nonlinear programming (MINLP) in which the macro and micro imbalance are penalized using absolute value and quadratic functions. We develop various reformulation-linearization techniques (RLTs) for the MINLP models, leading to three mathematical modelling variants: (i) a mixed-integer quadratically constrained program (MIQCP) and (ii) two mixed-integer programs (MIPs) for the absolute value penalty function and an MIQCP for the quadratic penalty function. Two novel exact techniques based on reformulation-decomposition techniques (RDTs) are developed to solve these models: a uni- and a bi-level logic-based Benders decomposition (LBBD). We motivate the LBBD methods with an application to BDORS in the University Health Network (UHN), consisting of three collaborating hospitals: Toronto General Hospital, Toronto Western Hospital, and Princess Margaret Cancer Centre in Toronto, Ontario, Canada. The uni-level LBBD method decomposes the model into a surgical suite location, OR allocation, and macro balancing master problem (MP) and micro OR balancing sub-problems (SPs) for each hospital-day. The bi-level approach uses a relaxed MP, consisting of a surgical suite location and relaxed allocation/macro balancing MP and two optimization SPs. The primary SP is formulated as a bin-packing problem to allocate patients to open operating rooms to minimize the number of ORs, while the secondary SP is the uni-level micro balancing SP. Using UHN datasets consisting of two datasets, hard MP/easy SPs and easy MP/hard SPs, we show that both LBBD approaches and both MIP models solved via Gurobi converge to ≈ 2% and ≈ 1–2% optimality gaps, on average, respectively, within 30 minutes runtime, whereas the MIQCP solved via Gurobi could not solve any instance of the UHN datasets given the same runtime. The uni- and bi-level LBBD approaches solved all instances of hard MP/easy SPs dataset to ≈ 11% and ≈ 2% optimality gaps, on average, respectively, within 30 minutes runtime, whereas MIQCP solved via Gurobi could not solve any of these instances. Additionally, we show that convergence of each LBBD varies depending on where in the decomposition the actual computational complexity lies.

Strong RLT1 bounds from decomposable Lagrangean relaxation for some quadratic 0–1 optimization problems with linear constraints

The Reformulation Linearization Technique (RLT) of Sherali and Adams (Manag Sci 32(10):1274–1290, 1986; SIAM J Discrete Math 3(3):411–430, 1990), when applied to a pure 0–1 quadratic optimization problem with linear constraints (P), constructs a hierarchy of LP (i.e., continuous and linear) models of increasing sizes. These provide monotonically improving continuous bounds on the optimal value of (P) as the level, i.e., the stage in the process, increases. When the level reaches the dimension of the original solution space, the last model provides an LP bound equal to the IP optimum. In practice, unfortunately, the problem size increases so rapidly that for large instances, even computing bounds for RLT models of level k (called RLTk) for small k may be challenging. Their size and their complexity increase drastically with k. To our knowledge, only results for bounds of levels 1, 2, and 3 have been reported in the literature. We are proposing, for certain quadratic problem types, a way of producing stronger bounds than continuous RLT1 bounds in a fraction of the time it would take to compute continuous RLT2 bounds. The approach consists in applying a specific decomposable Lagrangean relaxation to a specially constructed RLT1-type linear 0–1 model. If the overall Lagrangean problem does not have the integrality property, and if it can be solved as a 0–1 rather than a continuous problem, one may be able to obtain 0–1 RLT1 bounds of roughly the same quality as standard continuous RLT2 bounds, but in a fraction of the time and with much smaller storage requirements. If one actually decomposes the Lagrangean relaxation model, this two-step procedure, reformulation plus decomposed Lagrangean relaxation, will produce linear 0–1 Lagrangean subproblems with a dimension no larger than that of the original model. We first present numerical results for the Crossdock Door Assignment Problem, a special case of the Generalized Quadratic Assignment Problem. These show that just solving one Lagrangean relaxation problem in 0–1 variables produces a bound close to or better than the standard continuous RLT2 bound (when available) but much faster, especially for larger instances, even if one does not actually decompose the Lagrangean problem. We then present numerical results for the 0–1 quadratic knapsack problem, for which no RLT2 bounds are available to our knowledge, but we show that solving an initial Lagrangean relaxation of a specific 0–1 RLT1 decomposable model drastically improves the quality of the bounds. In both cases, solving the fully decomposed rather than the decomposable Lagrangean problem to optimality will make it feasible to compute such bounds for instances much too large for computing the standard continuous RLT2 bounds.

Risk-Averse Facility Location for Green Closed-Loop Supply Chain Networks Design under Uncertainty

With the increasing attention given to environmentalism, designing a green closed-loop supply chain network has been recognized as an important issue. In this paper, we consider the facility location problem, in order to reduce the total costs and CO2 emissions under an uncertain demand and emission rate. Particularly, we are more interested in the risk-averse method for providing more reliable solutions. To do this, we employ a coherent risk measure, conditional value-at-risk, to represent the underlying risk of uncertain demand and CO2 emission rate. The resulting optimization problem is a 0-1 mixed integer bi-objective programming, which is challenging to solve. We develop an improved reformulation-linearization technique, based on decomposed piecewise McCormick envelopes, to generate lower bounds efficiently. We show that the proposed risk-averse model can generate a more reliable solution than the risk-neutral model, both in reducing penalty costs and CO2 emissions. Moreover, the proposed algorithm outperforms and classic reformulation-linearization technique in convergence rate and gaps. Numerical experiments based on random data and a ‘real’ case are performed to demonstrate the performance of the proposed model and algorithm.

A Linear Solution Method of Generalized Robust Chance Constrained Real-Time Dispatch

In this letter, a novel solution method of generalized robust chance constrained real-time dispatch (GRCC-RTD) considering wind power uncertainty is proposed. GRCC models are advantageous in dealing with distributional uncertainty, however, they are difficult to solve because of the complex ambiguity set. By constructing traceable counterparts of the robust chance constraints and using the reformulation linearization technique, the model is equivalently transformed into a deterministic linear programming problem, which can be solved efficiently by off-the-shelf solvers. Numerical results verify the effectiveness and efficiency of the approach.

Tight-and-Cheap Conic Relaxation for the AC Optimal Power Flow Problem

The classical alternating current optimal power flow problem is highly nonconvex and generally hard to solve. Convex relaxations, in particular semidefinite, second-order cone, convex quadratic, and linear relaxations, have recently attracted significant interest. The semidefinite relaxation is the strongest among them and is exact for many cases. However, the computational efficiency for solving large-scale semidefinite optimization is lower than for second-order cone optimization. We propose a conic relaxation obtained by combining semidefinite optimization with the reformulation-linearization technique, commonly known as RLT. The proposed relaxation is stronger than the second-order cone relaxation and nearly as tight as the standard semidefinite relaxation. Computational experiments using standard test cases with up to 6515 buses show that the time to solve the new conic relaxation is up to one order of magnitude lower than for the chordal relaxation, a semidefinite relaxation technique that exploits the sparsity of power networks.

Investigating continuous power flow solutions of IEEE 14‐bus system

This letter focuses on the multiplicity of power flow (PF) equations and presents two continuous solutions for the widely studied IEEE 14‐bus system. The continuous solutions are located by a method combining the semidefinite program (SDP) relaxation and the reformulation linearization technique (RLT). Although the observation is nontrivial, it is of interest to researchers investigating the geometry or multiplicity nature of PF equations. © 2018 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

On linear programming relaxations for solving polynomial programming problems

This paper studies linear programming (LP) relaxations for solving polynomial programming problems. A polynomial programming problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP) by introducing new variables that represent nonlinear monomials and substituting them within the original formulation. Whereas Reformulation-Linearization Technique (RLT)-based LP relaxations constructed for the original formulation are tight, the relaxations generated using the equivalent lower degree formulations are smaller and require less computational effort to optimize in comparison to the effort the former requires. In this study, we analyze the strength and tractability of the standard RLT, the J-set, and recursive McCormick relaxations for polynomial programming problems, and identify the superior relaxation depending on the problem characteristics. Extensive computational results are provided to demonstrate the relative effectiveness of the standard RLT, J-set, and recursive McCormick algorithms using problems from the literature and randomly generated test instances.

Integrating combinatorial algorithms into a linear programming solver

While there are numerous linear (and nonlinear) solvers, as well as specialized algorithms for combinatorial problems, they are rarely used together. We wrote a new module for the XPRESS optimizer that lets us call the objects and functions of the LEMON C++ library. We tested this module by comparing two versions of the Dual Ascent Procedure (DAP) (Adams and Johnson in DIMACS Ser Discrete Math Theor Comput Sci 16:43–75, 1994). It is an iterative algorithm that produces lower bounds for the quadratic assignment problem (QAP). The DAP is based on the level-1 RLT-relaxation (reformulation-linearization technique) of the QAP. This formulation is significantly larger than the original QAP, but if we construct its Lagrangian dual, we can decompose the new relaxation to smaller linear assignment problems for a fixed Lagrange multiplier. The algorithm determines a better Lagrange multiplier in every iteration. We implemented two versions of the DAP with the XPRESS Optimization Software. In the first model we solved the smaller linear assignment problems with the linear programming methods of XPRESS, while in the second model we solved them by using the graph algorithms of the LEMON library via the new module. Using the instances of QAPLIB, the new module produced significantly better running times, which suggests that this direction of research might yield further results in the future.