Outer Approximation Method Research Articles

An extension of hybrid method without extrapolation step to equilibrium problems

In this paper, we introduce a new hybrid algorithm for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method and the hybrid (outer approximation) method. In this algorithm, only one optimization program is solved at each iteration without any extra-step dealing with the feasible set like as in the hybrid extragradient method and the hybrid Armijo linesearch method. A specially constructed half-space in the hybrid method is the reason for the absence of an optimization program in the proposed algorithm. The strongly convergent theorem is established and several numerical experiments are implemented to illustrate the convergence of the algorithm and compare it with others.

The Unit Commitment Problem With AC Optimal Power Flow Constraints

We propose a mathematical programming-based approach to optimize the unit commitment problem with alternating current optimal power flow (ACOPF) network constraints. This problem is a nonconvex mixed-integer nonlinear program (MINLP) that we solve through a solution technique based on the outer approximation method. Our solution technique cooptimizes real and reactive power scheduling and dispatch subject to both unit commitment constraints and ACOPF constraints. The proposed approach is a local solution method that leverages powerful linear and mixed-integer commercial solvers. We demonstrate the relative economic and operational impact of more accurate ACOPF constraint modeling on the unit commitment problem, when compared with copperplate and DCOPF constraint modeling approaches; we use a six-bus, the IEEE RTS-79, and the IEEE-118 test systems for this analysis. Our approach can be extended to solve larger scale power systems as well as include security constraints or uncertainty through decomposition techniques.

Cyclic subgradient extragradient methods for equilibrium problems

In this paper, we introduce a cyclic subgradient extragradient algorithm and its modified form for finding a solution of a system of equilibrium problems for a class of pseudomonotone and Lipschitz-type continuous bifunctions. The main idea of these algorithms originates from several previously known results for variational inequalities. The proposed algorithms are extensions of the subgradient extragradient method for variational inequalities to equilibrium problems and the hybrid (outer approximation) method. The paper can help in the design and analysis of practical algorithms and gives us a generalization of the most convex feasibility problems.

Stacking sequence and shape optimization of laminated composite plates via a level-set method

We consider the optimal design of composite laminates by allowing a variable stacking sequence and in-plane shape of each ply. In order to optimize both variables we rely on a decomposition technique which aggregates the constraints into one unique constraint margin function. Thanks to this approach, an exactly equivalent bi-level optimization problem is established. This problem is made up of an inner level represented by the combinatorial optimization of the stacking sequence and an outer level represented by the topology and geometry optimization of each ply. We propose for the stacking sequence optimization an outer approximation method which iteratively solves a set of mixed integer linear problems associated to the evaluation of the constraint margin function. For the topology optimization of each ply, we lean on the level set method for the description of the interfaces and the Hadamard method for boundary variations by means of the computation of the shape gradient. Numerical experiments are performed on an aeronautic test case where the weight is minimized subject to different mechanical constraints, namely compliance, reserve factor and buckling load.

Method of Outer Approximations and Adaptive Approximations for a Class of Matrix Games

We present a novel technique for obtaining global solutions to discrete min-max problems that arise naturally in the receding horizon control of unmanned craft in which the controls can be adjusted only in notches, e.g., stop, half forward, full forward, left or right $$60^{\circ }$$60?, and in the finite precision global solution of certain classes of semi-infinite min-max problems. The technique consists of a method for transcribing min-max problems over discrete sets into a matrix game and matrix game-specific adaptations of the Method of Outer Approximations and the Method of Adaptive Approximations, which are normally used for solving optimal control and semi-infinite min-max problems. The efficiency of the Method of Outer Approximations depends on having a good initial approximation to a solution. To this end, we make use of adaptive approximation techniques to decompose a large matrix game into a sequence of lower dimensional games, the solution of each giving rise to a very good initial approximation to a solution for the next game. We show that a basic approach for solving a min-max matrix game, in which one maximizes over the elements of columns and minimizes over the elements of the rows of a matrix, requires a number of function evaluations which is equal to the product of the number of rows and the number of columns of the matrix, while with our new approach our experimental results require only a number of function evaluations which is the product of the number rows and a number ranging from 2 to 20, shortening computing times from years to fractions of a minute.

Cutting planes for improved global logic-based outer-approximation for the synthesis of process networks

Abstract In this work, we present an improved global logic-based outer-approximation method (GLBOA) for the solution of nonconvex generalized disjunctive programs (GDP). The GLBOA allows the solution of nonconvex GDP models, and is particularly useful for optimizing the synthesis of process networks, which yields MINLP models that can be highly nonconvex. However, in many cases the NLP that results from fixing the discrete decisions is much simpler to solve than the original problem. The proposed method exploits this property. Two enhancements to the basic GLBOA are presented. The first enhancement seeks to obtain feasible solutions faster by dividing the basic algorithm into two stages. The first stage seeks to find feasible solutions faster by restricting the solution time of the problems and diversifying the search. The second stage guarantees the convergence by solving the original algorithm. The second enhancement seeks to tighten the lower bound of the algorithm by the use of cutting planes. The proposed method for obtaining cutting planes, the main contribution of this work, is a separation problem based on the convex hull of the feasible region of a subset of the constraints. Results and comparison with other global solvers show that the enhancements improve the performance of the algorithm, and that it is more effective in the tested problems at finding near optimal solutions compared to general-purpose global solvers.

Integrated design and control of semicontinuous distillation systems utilizing mixed integer dynamic optimization

Semicontinuous distillation systems are notoriously difficult to design and optimize because the structural parameters, operational parameters, and control system must all be determined simultaneously. In the past 15 years of research into semicontinuous systems, studies of the optimal design of these systems have all been limited in scope to small subsets of the parameters, which yields suboptimal and often unsatisfactory results. In this work, for the first time, the problem of integrated design and control of semicontinuous distillation processes is studied by using a mixed integer dynamic optimization (MIDO) problem formulation to optimize both the structural and control tuning parameters of the system. The public model library (PML) of gPROMS is used to simulate the process and the built-in optimization package of gPROMS is used to solve the MIDO via the deterministic outer approximation method. The optimization results are then compared to the heuristic particle swarm optimization (PSO) method.

Multi-Cuts Outer Approximation Method for Unit Commitment

This letter introduces a deterministic global optimization methods for unit commitment (UC) problem based on outer approximation method (OAM). The proposed Multi-cuts OAM (MCs-OAM) decomposes the UC problem into a mixed integer linear programming (MILP) master problem and several nonlinear programming (NLP) subproblems, whereas only one NLP in classic OAM. After elaborately designing the terminating criterion for solving the bigger but tighter MILP master problems, MCs-OAM can obtained higher quality solutions with fewer main iterations and less total CPU times, although solving more NLPs consumes more CPU times than OAM. The numerical results on 42 test systems of up to 200 units show that the MCs-OAM is very promising for large scale UC problems because that it can obtain high-quality solutions in reasonable time.

Multimedia Content Delivery in Millimeter Wave Home Networks

Millimeter wave (mm-wave) communication is a promising technology for short-range communications and high-speed services. It provides potential solutions for multimedia content delivery in home networks. However, approaches for resource allocation in traditional wireless networks may not be efficient for multimedia data transmissions in mm-wave networks. This is due to the very wide bandwidth and highly directional transmissions, which bring challenges as well as opportunities to the resource allocation of mm-wave transmissions. In this paper, we first characterize different usage scenarios of multimedia content delivery by introducing a set of utility functions. We then formulate a joint power and channel allocation problem based on a network utility maximization framework, which captures the spatial and frequency reuse of mm-wave communications. The formulated problem is a non-convex mixed integer programming (MIP) problem. We reformulate the problem into a convex MIP problem and propose a resource allocation algorithm based on outer approximation (OA) method. We further develop an efficient heuristic algorithm, which has a lower complexity than the OA based algorithm. Simulation results present the tradeoffs between the OA based and heuristic algorithms for different scenarios and show that our proposed algorithms substantially outperform recently proposed schemes in the literature.

Hybrid iterative algorithms for two families of finite maximal monotone mappings

In this paper, we introduce and analyze a new general hybrid iterative algorithm for finding a common element of the set of common zeros of two families of finite maximal monotone mappings, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping in a real Hilbert space. Our algorithm is based on four well-known methods: Mann’s iteration method, composite method, outer-approximation method and extragradient method. We prove the strong convergence theorem for the proposed algorithm. The results presented in this paper extend and improve the corresponding results of Wei and Tan (Fixed Point Theory Appl. 2014:77, 2014). Some special cases are also discussed.

Joint admission control and beamforming design for the interference cognitive radio network with partial channel state information case

In order to effectively utilise the spectrum of wireless network, secondary users (SUs) are often authorised to share spectrum with primary users (PUs) in cognitive networks. To guarantee effective transmission of PUs and SUs, the interference from the SUs to the PUs is limited and only the SUs that have the information rates (rates) than the corresponding thresholds are allowed to transmit their data. In order to select proper SUs to transmit and to maximise their achievable weighted sum rate (WSR) under rate requirement and interference constraint, the authors propose a joint admission control and beamforming design algorithm for partial channel state information (CSI) case. The WSR maximisation problem is first constructed by using approximation and convex relaxation, according to the statistical property of the CSI. Then a penalty factor is introduced to check the feasibility of such problem and instruct the admission of the SUs. Beyond the feasible SUs set, the WSR is maximised by updating the achievable rate of each user with rate profile technique and polyblock outer approximation method. Simulation results show that the proposed beamforming can achieve higher WSR than the beamforming of the existing baseline method.

An exact algorithm for the mean–standard deviation shortest path problem

This paper studies the reliable path problem in the form of minimizing the sum of mean and standard deviation of path travel time. For the case of independent link travel times, we show that the problem can be solved exactly by repeatedly solving a subproblem minimizing the sum of mean and variance of path travel time. The latter is an additive shortest path problem, and can be solved using a standard labeling algorithm. While these subproblems are similar in form to those obtained from Lagrangian relaxation, this formulation admits proof of finite convergence to the optimal solution. An iterative labeling algorithm is developed that solves the non-additive reliable path problem from a single origin to all destinations. Moreover, a labeling technique is employed to further reduce the computational time of the proposed algorithm by partially updating the network in each iteration. As an alternative, a bisection-type search algorithm is developed that solves the problem for the single-origin and single-destination case. Numerical tests are presented, indicating that the proposed algorithm outperform others recently proposed in the literature: unlike Lagrangian relaxation, two of the proposed algorithms find solutions exactly, and the computation time is an order of magnitude faster than outer approximation methods.

Reducing Generation Uncertainty by Integrating CSP With Wind Power: An Adaptive Robust Optimization-Based Analysis

The uncertainty of wind power generation brings problems in power system operation, such as requiring more reserves and possible frequency issues. In this paper, we propose an idea of combining concentrating solar power (CSP) plants with wind farms to reduce the overall uncertainty in the joint power output. Taking advantage of the dispatchability of CSP, the uncertainty of joint power generation is expected to decrease. Based on the operational model of CSP plants with thermal storage system, we search for the narrowest but robust bounds of the joint power output with a given uncertainty of the wind power output and solar power availability, and within operational constraints of CSP plants. The problem is formulated as an adaptive robust optimization (RO) problem, containing mixed-integer variables at the second stage. We introduce an algorithm that combines a nested column-and-constraint generation (C-CG) method and an outer approximation (OA) method to solve the problem. The case studies show that robust intervals for the joint power output can be obtained, and the obtained intervals can be significantly narrower than the original intervals of wind power.

An outer approximation method for an integration of supply chain network designing and assembly line balancing under uncertainty

In this paper a new model is proposed for the integrated problem of supply chain network designing and assembly line balancing under demand uncertainty. In this problem there are three types of entities: manufacturers, assemblers and customers. Manufacturers provide assemblers with components and assemblers use these components to produce the final products and satisfy the demand of the customers. This problem involves determining the location of manufacturers and assemblers in the network, balancing the assembly lines, and transportation of materials and products throughout the network. A mixed integer nonlinear programming formulation based on two stage stochastic programming method is developed to solve this problem to optimality. Moreover, an outer approximation (OA) method is proposed to efficiently solve this problem. The computational results show efficiency of the proposed OA method.

Outer Approximation Algorithm for One Class of Convex Mixed-Integer Nonlinear Programming Problems with Partial Differentiability

In this paper, we mainly study one convex mixed-integer nonlinear programming problem with partial differentiability and establish one outer approximation algorithm for solving this problem. With the help of subgradients, we use the outer approximation method to reformulate this convex problem as one equivalent mixed-integer linear program and construct an algorithm for finding optimal solutions. The result on finite steps convergence of the algorithm is also presented.

Convex mixed integer nonlinear programming problems and an outer approximation algorithm

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 paper generalizes and extends the outer approximation method in the sense of dealing with convex MINLPs from differentiable case to non-differentiable one.

On the Computation of Value Correspondences for Dynamic Games

Recursive game theory provides theoretic procedures for computing the equilibrium payoff or value sets of repeated games and the equilibrium payoff or value correspondences of dynamic games. In this paper, we propose and implement outer and inner approximation methods for equilibrium value correspondences that naturally occur in the analysis of dynamic games. The procedure utilizes set-valued step functions. We provide an application to a bilateral insurance game with storage.

Fast Gaussian kernel learning for classification tasks based on specially structured global optimization

For a practical pattern classification task solved by kernel methods, the computing time is mainly spent on kernel learning (or training). However, the current kernel learning approaches are based on local optimization techniques, and hard to have good time performances, especially for large datasets. Thus the existing algorithms cannot be easily extended to large-scale tasks. In this paper, we present a fast Gaussian kernel learning method by solving a specially structured global optimization (SSGO) problem. We optimize the Gaussian kernel function by using the formulated kernel target alignment criterion, which is a difference of increasing (d.i.) functions. Through using a power-transformation based convexification method, the objective criterion can be represented as a difference of convex (d.c.) functions with a fixed power-transformation parameter. And the objective programming problem can then be converted to a SSGO problem: globally minimizing a concave function over a convex set. The SSGO problem is classical and has good solvability. Thus, to find the global optimal solution efficiently, we can adopt the improved Hoffman’s outer approximation method, which need not repeat the searching procedure with different starting points to locate the best local minimum. Also, the proposed method can be proven to converge to the global solution for any classification task. We evaluate the proposed method on twenty benchmark datasets, and compare it with four other Gaussian kernel learning methods. Experimental results show that the proposed method stably achieves both good time-efficiency performance and good classification performance.

A Novel Separable Model and Decomposition Method for Sensor Locational Decision Problem

This paper proposes a new separable model for the sensor locational decision problem covering a line (SLDPCL). By decomposing a multivariate function into several univariate functions, a separable outer approximation methodology that can be used to improve the outer approximation of classical convex programming technique is presented. A novel outer approximation method (OAM) for this proposed separable model is proposed. The algorithm alternates between solving a mixed integer linear program and a convex nonlinear program (NLP). An improved interior point method based on optimal centering parameter is employed to solve the NLP subproblem. The simulation results for test instances that range in size from 10 to 20000 sensors show that the proposed method is fast and robust, and the method is very promising for large-scale SLDPCL problems due to its excellent scalability.

Outer Approximation and Outer-Inner Approximation Approaches for Unit Commitment Problem

This paper proposes a new separable model for the unit commitment (UC) problem and three deterministic global optimization methods for it ensuring convergence to the global optimum within a desired tolerance. By decomposing a multivariate function into several univariate functions, a tighter outer approximation methodology that can be used to improve the outer approximations of several classical convex programming techniques is presented. Based on the idea of the outer approximation (OA) method and the proposed separable model, an outer-inner approximation (OIA) approach is also presented to solve this new formulation of UC problem. In this OIA approach, the UC problem is decomposed into a tighter outer approximation subproblem and an inner approximation subproblem, where the former leads to a better lower bound than the OA method, and the later provides a better upper bound. The simulation results for systems of up to 100 units with 24 h are compared with those of previously published methods, which show that the OIA approach is very promising due to the excellent performance. The proposed approaches are also applied to the large-scale systems of up to 1000 units with 24 h, and systems of up to 100 units with 96 h and 168 h.