Bilinear Programming Research Articles

A practicable branch and bound algorithm for sum of linear ratios problem

Abstract This article presents a practicable algorithm for globally solving sum of linear ratios problem (SLR). The algorithm works by globally solving a bilinear programming problem (EQ) that is equivalent to the problem (SLR). In the algorithm, by utilizing convex envelope and concave envelope of bilinear function, the initial nonconvex programming problem is reduced to a sequence of linear relaxation programming problems. In order to improve the computational efficiency of the algorithm, a new accelerating technique is introduced, which provides a theoretical possibility to delete a large part of the investigated region in which there exists no global optimal solution of the (EQ). By combining this innovative technique with branch and bound operations, a global optimization algorithm is designed for solving the problem (SLR). Finally, numerical experimental results show the feasibility and efficiency of the proposed algorithm.

Coping with uncertainties in production planning through fuzzy mathematical programming: application to steel rolling industry

This paper adopts the approach of fuzzy set theory into the context of a practical production planning problem encountered frequently in steel rolling mills, where the objective is to establish a cost–minimising master production schedule. To better capture the uncertainties associated with the market demand, the problem is formulated as a fuzzy mixed integer bilinear program (FMIBLP) in which the demand constraints are assumed to be rather flexible and characterised by triangular membership functions. The aspiration level for the decision maker is represented by a linear function where the tolerance limits for this function are determined based on the degree of flexibility in demand that the decision maker is willing to undertake. The fuzzy decision set is obtained using two different types of aggregators which, in turn, allows for the transformation of the fuzzy model into a crisp one seeking the maximum value for the aspiration level. A linearisation scheme is first adopted to transform the bilinear model into an equivalent linear model and then an exterior penalty function based algorithm is employed to the linearised version in order to obtain 'near optimal' solutions that minimise deviations from integral batches. Computational experiments are carried out for different problem instances under both aggregation operators and the results are reported.

Minimum norm solution of the absolute value equations via simulated annealing algorithm

This paper mainly aims to obtain a minimum norm solution to the NP-hard absolute value equations (AVE) \(Ax-|x|=b\). In our proposed method, first we show that the AVE is equivalent to a bilinear programming problem and then we present a system tantamount to this problem. To find the minimum norm solution to the AVE, we solve a quadratic programming problem with quadratic and linear constraints. Eventually, we solve the obtained programming problem by Simulated Annealing algorithm. Numerical results support the efficiency of the proposed method and the high accuracy of calculation.

Task assignment with controlled and autonomous agents

We analyse assignment problems in which not every agent is controlled by the central planner. The autonomous agents search for vacant tasks guided by their own preference orders over available tasks. The goal of the central planner is to maximise the total value of the assignment, taking into account the behaviour of the uncontrolled agents. Such optimisation problems arise in numerous real-world situations, ranging from organisational economics to “crowdsourcing” and disaster response. We show that the problem faced by the central planner can be transformed into a mixed integer bilevel optimisation problem. Then we demonstrate how this program can be reduced to a disjoint bilinear program, which is much more manageable computationally.

Absolute Value Constraint Based Method for Interval Optimization to SCED Model

This letter proposes an absolute value based method to solve the interval linear optimization problem with application to security constrained economic dispatch (SCED). To avoid expensive computation associated with solving combinatorial linear programming (LP) problems for interval upper bound, firstly a bilinear programming model is formulated using duality theory. The equality constraints are then converted to absolute value constraints. Lastly, the absolute value operator is eliminated through introducing two nonnegative slack variables and complementary slackness condition. The resulting new bilinear programming model can be effectively solved by the branch and bound method with linear relaxation technique to obtain the global optimal solution. Numerical results demonstrate the effectiveness of the proposed method in improving solution and computation time.

Mathematical Programming Formulations and Algorithms for Discrete k-Median Clustering of Time-Series Data

Discrete k-median (DKM) clustering problems arise in many real-life applications that involve time-series data sets, in which nondiscrete clustering methods may not represent the problem domain adequately. In this study, we propose mathematical programming formulations and solution methods to efficiently solve the DKM clustering problem. We develop approximation algorithms from a bilinear formulation of the discrete k-median problem using an uncoupled bilinear program algorithm. This approximation algorithm, which we refer to as DKM-L, is composed of two alternating linear programs, where one can be solved in linear time and the other is a minimum cost assignment problem. We then modify this algorithm by replacing the assignment problem with an efficient sequential algorithm for a faster approximation, which we call DKM-S. We also propose a compact exact integer formulation, DKM-I, and a more efficient network design-based exact mixed-integer formulation, DKM-M. All of our methods use arbitrary pairwise distance matrices as input. We apply our methods to simulated single-variate and multivariate random walk time-series data. We report comparative clustering performances using normalized mutual information (NMI) and solution speeds among the DKM methods we propose. We also compare our methods to other clustering algorithms that can operate with distance matrices, such as hierarchical cluster trees (HCT) and partition around medoids (PAM). We present NMI scores and classification accuracies of our DKM algorithms compared to HCT and PAM using five different distance measures on simluated data, as well as public benchmark and real-life neural time-series data sets. We show that DKM-S is much faster than HCT, PAM, and all other DKM methods and produces consistently good clustering results on all data sets.

A Probabilistic Framework for Reserve Scheduling and ${\rm N}-1$ Security Assessment of Systems With High Wind Power Penetration

We propose a probabilistic framework to design an N-1 secure day-ahead dispatch and determine the minimum cost reserves for power systems with wind power generation. We also identify a reserve strategy according to which we deploy the reserves in real-time operation, which serves as a corrective control action. To achieve this, we formulate a stochastic optimization program with chance constraints, which encode the probability of satisfying the transmission capacity constraints of the lines and the generation limits. To incorporate a reserve decision scheme, we take into account the steady-state behavior of the secondary frequency controller and, hence, consider the deployed reserves to be a linear function of the total generation-load mismatch. The overall problem results in a chance constrained bilinear program. To achieve tractability, we propose a convex reformulation and a heuristic algorithm, whereas to deal with the chance constraint we use a scenario-based-approach and an approach that considers only the quantiles of the stationary distribution of the wind power error. To quantify the effectiveness of the proposed methodologies and compare them in terms of cost and performance, we use the IEEE 30-bus network and carry out Monte Carlo simulations, corresponding to different wind power realizations generated by a Markov chain-based model.

Buyer's quantile hedge portfolios in discrete-time trading

The problem of quantile hedging for American claims is studied from the perspective of the buyer of a contingent claim by minimizing the ‘expected failure ratio’. After a general study of the problem in infinite-state spaces, we pass to finite dimensions and examine the properties of the resulting finite-dimensional optimization problems. In finite-state probability spaces we obtain a bilinear programming formulation that admits an exact linearization using binary exercise variables. Numerical results with S&P 500 index options demonstrate the computational viability of the formulations.

Convexification method for bilinear transmission expansion problem

SUMMARY This research presents a method for convexifying the non-linear constraints and the objective function for the Transmission Network Expansion Planning Problem (TNEP). The TNEP seeks to identify the best set of transmission capacity additions to meet a future electric power demand. The TNEP is a non-convex Mixed Integer Non-linear Programming problem for which a variety of primal solution methods have been designed. In this paper, the TNEP is formulated as a bilinear programming problem subject to bilinear constraints. This allows resolving the non-linearities through convexification of the non-linear objective function and the constraints for a relaxation of the TNEP in which integrality requirements are removed. The final solution for the original TNEP problem is obtained by using a branch and bound strategy. It was found by using a two-node test case that the convexification method presented in this paper in conjunction with a brand and bound strategy identifies the optimal solution. Successfully solving a small-scale test problem through convexification presents a promising scenario to solve larger scale problems. This primal-dual method allows identifying the duality gap which is a limitation of primal methods. Further multidisciplinary research is needed to extend the method presented here to larger cases and real life networks. Copyright © 2013 John Wiley & Sons, Ltd.

Global optimization of bilinear programs with a multiparametric disaggregation technique

In this paper, we present the derivation of the multiparametric disaggregation technique (MDT) by Teles et al. (J. Glob. Optim., 2011) for solving nonconvex bilinear programs. Both upper and lower bounding formulations corresponding to mixed-integer linear programs are derived using disjunctive programming and exact linearizations, and incorporated into two global optimization algorithms that are used to solve bilinear programming problems. The relaxation derived using the MDT is shown to scale much more favorably than the relaxation that relies on piecewise McCormick envelopes, yielding smaller mixed-integer problems and faster solution times for similar optimality gaps. The proposed relaxation also compares well with general global optimization solvers on large problems.

A Difference-Index Based Ranking Bilinear Programming Approach to Solving Bimatrix Games with Payoffs of Trapezoidal Intuitionistic Fuzzy Numbers

The aim of this paper is to develop a bilinear programming method for solving bimatrix games in which the payoffs are expressed with trapezoidal intuitionistic fuzzy numbers (TrIFNs), which are called TrIFN bimatrix games for short. In this method, we define the value index and ambiguity index for a TrIFN and propose a new order relation of TrIFNs based on the difference index of value index to ambiguity index, which is proven to be a total order relation. Hereby, we introduce the concepts of solutions of TrIFN bimatrix games and parametric bimatrix games. It is proven that any TrIFN bimatrix game has at least one satisfying Nash equilibrium solution, which is equivalent to the Nash equilibrium solution of corresponding parametric bimatrix game. The latter can be obtained through solving the auxiliary parametric bilinear programming model. The method proposed in this paper is demonstrated with a real example of the commerce retailers’ strategy choice problem.

A Global Optimization Algorithm for Sum of Linear Ratios Problem

We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

Oops! I cannot do it again: Testing for recursive feasibility in MPC

One of the most fundamental problems in model predictive control (MPC) is the lack of guaranteed stability and feasibility. It is shown how Farkas’ Lemma in combination with bilevel programming and disjoint bilinear programming can be used to search for problematic initial states which lack recursive feasibility, thus invalidating a particular MPC controller. Alternatively, the method can be used to derive a certificate that the problem is recursively feasible. The results are initially derived for nominal linear MPC, and thereafter extended to the additive disturbance case.

Lower hedging of American contingent claims with minimal surplus risk in finite-state financial markets by mixed-integer linear programming

The lower hedging problem with a minimal expected surplus risk criterion in incomplete markets is studied for American claims in finite state financial markets. It is shown that the lower hedging problem with linear expected surplus criterion for American contingent claims in finite state markets gives rise to a non-convex bilinear programming formulation which admits an exact linearization. The resulting mixed-integer linear program can be readily processed by available software.

Decomposition strategy for the stochastic pooling problem

The stochastic pooling problem is a type of stochastic mixed-integer bilinear program arising in the integrated design and operation of various important industrial networks, such as gasoline blending, natural gas production and transportation, water treatment, etc. This paper presents a rigorous decomposition method for the stochastic pooling problem, which guarantees finding an \({\epsilon}\) -optimal solution with a finite number of iterations. By convexification of the bilinear terms, the stochastic pooling problem is relaxed into a lower bounding problem that is a potentially large-scale mixed-integer linear program (MILP). Solution of this lower bounding problem is then decomposed into a sequence of relaxed master problems, which are MILPs with much smaller sizes, and primal bounding problems, which are linear programs. The solutions of the relaxed master problems yield a sequence of nondecreasing lower bounds on the optimal objective value, and they also generate a sequence of integer realizations defining the primal problems which yield a sequence of nonincreasing upper bounds on the optimal objective value. The decomposition algorithm terminates finitely when the lower and upper bounds coincide (or are close enough), or infeasibility of the problem is indicated. Case studies involving two example problems and two industrial problems demonstrate the dramatic computational advantage of the proposed decomposition method over both a state-of-the-art branch-and-reduce global optimization method and explicit enumeration of integer realizations, particularly for large-scale problems.

Comments on: Algorithms for linear programming with linear complementarity constraints

This paper addresses the solution of linear programming problems that further incorporate complementarity restrictions involving designated pairs of complementary variables, as well as its generalization where the objective function is a nonlinear continuously differentiable function. Such problems subsume mixed linear complementary problems, absolute value programs, the linear bilevel programming problem, and the bilinear programming problem as well as more general nonconvex quadratic programs, among many others. The existing research on these problems focuses on finding strongly stationary points, or more general B-stationary points, both of which are necessary characterizations for local minima. However, computing B-stationary points is more combinatorial since it involves characterizing solutions that turn out to be Karush–Kuhn–Tucker (KKT) points for an entire family of nonlinear programs. In special nondegenerate cases (where at least one of each complementary pair is nonzero in any complementary solution) or under a special linear independence constraint qualification, B-stationary points also turn out to be strongly stationary. Among several algorithms for computing such strongly stationary or B-stationary points, the most competitive method to-date is a complementary active-set algorithm (CASET) and its advanced variants. In several studies, it has been verified that this procedure works well, even for degenerate problems, outperforming other nonlinear programming approaches such as penalty methods, non-smooth optimization techniques, interior point methods, and sequential quadratic programming algorithms. The CASET algorithm requires an initial complementary feasible solution to start with, which can always be computed by using an enumerative method.

A bilinear programming model and a modified branch-and-bound algorithm for production planning in steel rolling mills with substitutable demand

In this paper, we address an instance of the dynamic capacitated multi-item lot-sizing problem (CMILSP) typically encountered in steel rolling mills. Production planning is carried out at the master production schedule level, where the various end items lot sizes are determined such that the total cost is minimised. Through incorporating the various technological constraints associated with the manufacturing process, the integrated production–inventory problem is formulated as a mixed integer bilinear program (MIBLP). Typically, such class of mathematical models is solved via linearisation techniques which transform the model to an equivalent MILP (mixed integer linear program) at the expense of increased model dimensionality. This paper presents an alternative branch-and-bound based algorithm that exploits the special structure of the mathematical model to minimise the number of branches and obtain the bound at each node. The performance of our algorithm is benchmarked against that of a classical linearisation technique for several problem instances and the obtained results are reported.

Primal-dual bilinear programming solution of the absolute value equation

We propose a finitely terminating primal-dual bilinear programming algorithm for the solution of the NP-hard absolute value equation (AVE): Ax − |x| = b, where A is an n × n square matrix. The algorithm, which makes no assumptions on AVE other than solvability, consists of a finite number of linear programs terminating at a solution of the AVE or at a stationary point of the bilinear program. The proposed algorithm was tested on 500 consecutively generated random instances of the AVE with n = 10, 50, 100, 500 and 1,000. The algorithm solved 88.6% of the test problems to an accuracy of 10−6.

PRICING SWING OPTIONS WITH TYPICAL CONSTRAINTS

We propose a pricing method by mathematical programming for swing options with typical constraints on a lattice model. We show that the problem of pricing typical swing options has a particular optimal solution such that there are only seven kinds of changed amounts in the solution. Using the solution, we formulate the pricing problem as a linear program. The solution can be applied to the methods of Jaillet, Ronn and Tompaidis (2004), and Barrera-Esteve et al. (2006) for improving time complexity. Another feature of our method is the capability to price swing options in an incomplete market. In an incomplete market, the price of a swing option is defined as an upper and a lower bound of arbitrage-free prices. We formulate the problem of finding an upper bound as a linear program. For a lower bound, we give a bilinear programming formulation.

Approaching minimum time control of timed continuous Petri nets

This paper considers the problem of controlling timed continuous Petri nets under infinite server semantics. The proposed control strategy assigns piecewise constant flows to transitions in order to reach the target state. First, by using linear programming, a method driving the system from the initial to the target state through a linear trajectory is developed. Then, in order to improve the time of the trajectory, intermediate states are added by means of bilinear programming. Finally, in order to handle potential perturbations, we develop a closed loop control strategy that follows the trajectory computed by the open loop control by calculating a new state after each time step. The algorithms developed here are applied to a manufacturing system.