Showing papers on "Nonlinear programming published in 2014"

Introductory Lectures on Convex Optimization: A Basic Course

Abstract: It was in the middle of the 1980s, when the seminal paper by Kar- markar opened a new epoch in nonlinear optimization The importance of this paper, containing a new polynomial-time algorithm for linear op- timization problems, was not only in its complexity bound At that time, the most surprising feature of this algorithm was that the theoretical pre- diction of its high efficiency was supported by excellent computational results This unusual fact dramatically changed the style and direc- tions of the research in nonlinear optimization Thereafter it became more and more common that the new methods were provided with a complexity analysis, which was considered a better justification of their efficiency than computational experiments In a new rapidly develop- ing field, which got the name "polynomial-time interior-point methods", such a justification was obligatory Afteralmost fifteen years of intensive research, the main results of this development started to appear in monographs [12, 14, 16, 17, 18, 19] Approximately at that time the author was asked to prepare a new course on nonlinear optimization for graduate students The idea was to create a course which would reflect the new developments in the field Actually, this was a major challenge At the time only the theory of interior-point methods for linear optimization was polished enough to be explained to students The general theory of self-concordant functions had appeared in print only once in the form of research monograph [12]

GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear Programming

Abstract: A general-purpose MATLAB software program called GPOPS--II is described for solving multiple-phase optimal control problems using variable-order Gaussian quadrature collocation methods. The software employs a Legendre-Gauss-Radau quadrature orthogonal collocation method where the continuous-time optimal control problem is transcribed to a large sparse nonlinear programming problem (NLP). An adaptive mesh refinement method is implemented that determines the number of mesh intervals and the degree of the approximating polynomial within each mesh interval to achieve a specified accuracy. The software can be interfaced with either quasi-Newton (first derivative) or Newton (second derivative) NLP solvers, and all derivatives required by the NLP solver are approximated using sparse finite-differencing of the optimal control problem functions. The key components of the software are described in detail and the utility of the software is demonstrated on five optimal control problems of varying complexity. The software described in this article provides researchers a useful platform upon which to solve a wide variety of complex constrained optimal control problems.

Manopt, a matlab toolbox for optimization on manifolds

Abstract: Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. By dealing internally with most of the differential geometry, the package aims particularly at lowering the entrance barrier.

A Collection of Test Problems for Constrained Global Optimization Algorithms

Abstract: Quadratic programming test problems.- Quadratically constrained test problems.- Nonlinear programming test problems.- Distillation column sequencing test problems.- Pooling/blending test problems.- Heat exchanger network synthesis test problems.- Phase and chemical reaction equilibrium test problems.- Comlpex chemical reactor network test problems.- Reactor-seperator-recycle system test problems.- Mechanical design test problems.- VLSI design test problems.

An efficient 3D topology optimization code written in Matlab

Abstract: This paper presents an efficient and compact Matlab code to solve three-dimensional topology optimization problems. The 169 lines comprising this code include finite element analysis, sensitivity analysis, density filter, optimality criterion optimizer, and display of results. The basic code solves minimum compliance problems. A systematic approach is presented to easily modify the definition of supports and external loads. The paper also includes instructions to define multiple load cases, active and passive elements, continuation strategy, synthesis of compliant mechanisms, and heat conduction problems, as well as the theoretical and numerical elements to implement general non-linear programming strategies such as SQP and MMA. The code is intended for students and newcomers in the topology optimization. The complete code is provided in Appendix C and it can be downloaded from http://top3dapp.com .

Solving Nonconvex Optimal Control Problems by Convex Optimization

Abstract: Motivated by aerospace applications, this paper presents a methodology to use second-order cone programming to solve nonconvex optimal control problems The nonconvexity arises from the presence of concave state inequality constraints and nonlinear terminal equality constraints The development relies on a solution paradigm, in which the concave inequality constraints are approximated by successive linearization Analysis is performed to establish the guaranteed satisfaction of the original inequality constraints, the existence of the successive solutions, and the equivalence of the solution of the original problem to the converged successive solution These results lead to a rigorous proof of the convergence of the successive solutions under appropriate conditions as well as nonconservativeness of the converged solution The nonlinear equality constraints are treated in a two-step procedure in which the constraints are first approximated by first-order expansions, then compensated by second-order correct

Forest Optimization Algorithm

Abstract: In this article, a new evolutionary algorithm, Forest Optimization Algorithm (FOA), suitable for continuous nonlinear optimization problems has been proposed. It is inspired by few trees in the forests which can survive for several decades, while other trees could live for a limited period. In FOA, seeding procedure of the trees is simulated so that, some seeds fall just under the trees, while others are distributed in wide areas by natural procedures and the animals that feed on the seeds or fruits. Application of the proposed algorithm on some benchmark functions demonstrated its good capability in comparison with Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). Also we tested the performance of FOA on feature weighting as a real optimization problem and the results of the experiments showed the good performance of FOA in some data sets from the UCI repository.

Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Methods

Abstract: This work presents a review of the main deterministic mixed-integer nonlinear programming (MINLP) solution methods for problems with convex and nonconvex functions. An overview for deriving MINLP formulations through generalized disjunctive programming (GDP), which is an alternative higher-level representation of MINLP problems, is also presented. A review of solution methods for GDP problems is provided. Some relevant applications of MINLP and GDP in process systems engineering are described in this work.

A derivative-free methodology with local and global search for the constrained joint optimization of well locations and controls

Abstract: In oil field development, the optimal location for a new well depends on how it is to be operated. Thus, it is generally suboptimal to treat the well location and well control optimization problems separately. Rather, they should be considered simultaneously as a joint problem. In this work, we present noninvasive, derivative-free, easily parallelizable procedures to solve this joint optimization problem. Specifically, we consider Particle Swarm Optimization (PSO), a global stochastic search algorithm; Mesh Adaptive Direct Search (MADS), a local search procedure; and a hybrid PSO–MADS technique that combines the advantages of both methods. Nonlinear constraints are handled through use of filter-based treatments that seek to minimize both the objective function and constraint violation. We also introduce a formulation to determine the optimal number of wells, in addition to their locations and controls, by associating a binary variable (drill/do not drill) with each well. Example cases of varying complexity, which include bound constraints, nonlinear constraints, and the determination of the number of wells, are presented. The PSO–MADS hybrid procedure is shown to consistently outperform both stand-alone PSO and MADS when solving the joint problem. The joint approach is also observed to provide superior performance relative to a sequential procedure.

Global optimization for sustainable design and synthesis of algae processing network for CO2 mitigation and biofuel production using life cycle optimization

Abstract: Global optimization for sustainable design and synthesis of a large-scale algae processing network under economic and environmental criteria is addressed. An algae processing network superstructure including 7800 processing routes is proposed. Based on the superstructure, a multiobjective mixed-integer nonlinear programming (MINLP) model is developed to simultaneously optimize the unit cost and the unit global warming potential (GWP). To efficiently solve the nonconvex MINLP model with separable concave terms and mixed-integer fractional terms in the objective functions, a global optimization strategy that integrates a branch-and-refine algorithm based on successive piecewise linear approximations is proposed and an exact parametric algorithm based on Newton’s method. Two Pareto-optimal curves are obtained for biofuel production and biological carbon sequestration, respectively. The unit annual biofuel production cost ranges from $7.02/gasoline gallon equivalent (GGE) to $9.71/GGE, corresponding to unit GWP’s of 26.491 to 16.52 kg CO2-eq/GGE, respectively. V C 2014 American Institute of Chemical Engineers AIChE J, 60: 3195–3210, 2014

MEIGO: an open-source software suite based on metaheuristics for global optimization in systems biology and bioinformatics

Abstract: Optimization is the key to solving many problems in computational biology. Global optimization methods, which provide a robust methodology, and metaheuristics in particular have proven to be the most efficient methods for many applications. Despite their utility, there is a limited availability of metaheuristic tools. We present MEIGO, an R and Matlab optimization toolbox (also available in Python via a wrapper of the R version), that implements metaheuristics capable of solving diverse problems arising in systems biology and bioinformatics. The toolbox includes the enhanced scatter search method (eSS) for continuous nonlinear programming (cNLP) and mixed-integer programming (MINLP) problems, and variable neighborhood search (VNS) for Integer Programming (IP) problems. Additionally, the R version includes BayesFit for parameter estimation by Bayesian inference. The eSS and VNS methods can be run on a single-thread or in parallel using a cooperative strategy. The code is supplied under GPLv3 and is available at http://www.iim.csic.es/~gingproc/meigo.html . Documentation and examples are included. The R package has been submitted to BioConductor. We evaluate MEIGO against optimization benchmarks, and illustrate its applicability to a series of case studies in bioinformatics and systems biology where it outperforms other state-of-the-art methods. MEIGO provides a free, open-source platform for optimization that can be applied to multiple domains of systems biology and bioinformatics. It includes efficient state of the art metaheuristics, and its open and modular structure allows the addition of further methods.

Accommodating High Penetrations of PEVs and Renewable DG Considering Uncertainties in Distribution Systems

Abstract: This paper proposes a multi-year multi-objective planning algorithm for enabling distribution networks to accommodate high penetrations of plug-in electric vehicles (PEVs) in conjunction with renewable distributed generation (DG). The proposed algorithm includes consideration of uncertainties and will help local distribution companies (LDC) better assess the expected impacts of PEVs on their networks and on proposed renewable DG connections. The goal of the proposed algorithm is to minimize greenhouse gas emissions and system costs during the planning horizon. An approach based on a non-dominated sorting genetic algorithm (NDSGA) is utilized to solve the planning problem of determining the optimal level of PEV penetration as well as the location, size, and year of installation of renewable DG units. The planning problem is defined in terms of multi-objective mixed integer nonlinear programming. The outcomes of the planning problem represent the Pareto frontier, which describes the optimal system solutions, from which the LDC can choose the system operating point, based on its preferences.

Optimal control for a Formula One car with variable parameters

Abstract: The minimum-lap-time optimal control problem for a Formula One race car is solved using direct transcription and nonlinear programming. Features of this work include significantly reduced full-lap solution times and the simultaneous optimisation of the driven line, the driver controls and multiple car set-up parameters. It is shown that significant reductions in the driven lap time can be obtained from track-specific set-up parameter optimisation. Reduced computing times are achieved using a combination of a track description based on curvilinear coordinates, analytical derivatives and model non-dimensionalisation. The curvature of the track centre line is found by solving an auxiliary optimal control problem that negates the difficulties associated with integration drift and trajectory closure.

Adaptive Penalty-Based Distributed Stochastic Convex Optimization

Abstract: In this work, we study the task of distributed optimization over a network of learners in which each learner possesses a convex cost function, a set of affine equality constraints, and a set of convex inequality constraints. We propose a fully distributed adaptive diffusion algorithm based on penalty methods that allows the network to cooperatively optimize the global cost function, which is defined as the sum of the individual costs over the network, subject to all constraints. We show that when small constant step-sizes are employed, the expected distance between the optimal solution vector and that obtained at each node in the network can be made arbitrarily small. Two distinguishing features of the proposed solution relative to other approaches is that the developed strategy does not require the use of projections and is able to track drifts in the location of the minimizer due to changes in the constraints or in the aggregate cost itself. The proposed strategy is able to cope with changing network topology, is robust to network disruptions, and does not require global information or rely on central processors.

Exactness of Semidefinite Relaxations for Nonlinear Optimization Problems with Underlying Graph Structure

Abstract: This work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems, including quadratic and polynomial optimization problems. The main objective of this paper is to investigate how the (hidden) structure of a given real/complex-valued optimization problem makes it easy to solve. To this end, three conic relaxations are proposed. Necessary and sufficient conditions are derived for the exactness of each of these relaxations, and it is shown that these conditions are satisfied if the optimization problem is highly structured. More precisely, the structure of the optimization problem is mapped into a generalized weighted graph, where each edge is associated with a weight set extracted from the coefficients of the optimization problem. In the real-valued case, it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph. It is also proved that if some of these conditi...

AP Association for Proportional Fairness in Multirate WLANs

Abstract: In this paper, we investigate the problem of achieving proportional fairness via access point (AP) association in multirate WLANs. This problem is formulated as a nonlinear programming with an objective function of maximizing the total user bandwidth utilities in the whole network. Such a formulation jointly considers fairness and AP selection. We first propose a centralized algorithm Non-Linear Approximation Optimization for Proportional Fairness (NLAO-PF) to derive the user-AP association via relaxation. Since the relaxation may cause a large integrality gap, a compensation function is introduced to ensure that our algorithm can achieve at least half of the optimal in the worst case. This algorithm is assumed to be adopted periodically for resource management. To handle the case of dynamic user membership, we propose a distributed heuristic Best Performance First (BPF) based on a novel performance revenue function, which provides an AP selection criterion for newcomers. When an existing user leaves the network, the transmission times of other users associated with the same AP can be redistributed easily based on NLAO-PF. Extensive simulation study has been performed to validate our design and to compare the performance of our algorithms to those of the state of the art.

Multiobjective multistage distribution system planning using tabu search

Abstract: This study presents a multiobjective tabu search algorithm to solve the multistage planning problem of a distribution system formulated as a multiobjective dynamic mixed integer non-linear programming problem. Multiobjective problems do not have a specific solution, but a set of solutions that allows us to observe the trade-off among the analysed objectives. Taking into account this concept, the objective functions of the model proposed in this study are: costs (investment and operational) and reliability. The actions deemed in this model for each period of the planning horizon are: increase in the capacity of existing substations (or construction of new ones), exchange of cables in existing lines (and construction of new feeders), reconfiguration of the network, allocation of sectionalising switches and construction of tie lines. The system's reliability is evaluated by means of the non-supplied energy under contingencies using the n − 1 criterion. By line switching and the use of tie lines, part of the loads affected by a contingency can be restored, thus, the non-supplied energy can be evaluated by solving a distribution network restoration problem. Numerical results are presented for a 54-bus system.

Energy-Efficient Train Operation in Urban Rail Transit Using Real-Time Traffic Information

Abstract: Energy-efficient train operation represents an important issue for daily operational urban rail transit. Most energy-efficient train operation strategies are normally planned according to a timetable, which is designed by offline traffic information. In this paper, a new energy-efficient train operation model based on real-time traffic information is proposed from the geometric and topographic points of view through a nonlinear programming method, leading to an energy-efficient driving strategy with real-time interstation running time monitored by the automatic train supervision system. The novelty of this work lies not only in the establishment of a new model for energy-efficient train operation but also in the utilization of combining analytical and numerical methods for deriving energy-efficient train operation strategies. More specifically, the energy-efficient operation model is built based on trajectory analysis when the energy-efficient optimal controls are applied, from which an energy-efficient reference trajectory is obtained under the running time and distance constraints, in which the nonlinear programming method is utilized. In contrast to most existing methods, the proposed model turns out to be a small-scale problem, and the difficulties of solving partial differential equations or the process of predetermining and reiteratively calculating some key factors as traditionally involved are avoided. Thus, it is more feasible to implement the strategy and easier to make real-time adjustment if needed. The comparative analysis and the simulation verification with the actual operating data confirm the effectiveness of the proposed method. With the proposed method, some delayed trains are able to maintain punctuality at the next station and sometimes even reducing energy consumption.

Computationally Efficient Model Predictive Control Algorithms: A Neural Network Approach

Abstract: This book thoroughly discusses computationally efficient (suboptimal) Model Predictive Control (MPC) techniques based on neural models The subjects treated include: A few types of suboptimal MPC algorithms in which a linear approximation of the model or of the predicted trajectory is successively calculated on-line and used for prediction Implementation details of the MPC algorithms for feed forward perceptron neural models, neural Hammerstein models, neural Wiener models and state-space neural models The MPC algorithms based on neural multi-models (inspired by the idea of predictive control) The MPC algorithms with neural approximation with no on-line linearization The MPC algorithms with guaranteed stability and robustness Cooperation between the MPC algorithms and set-point optimization Thanks to linearization (or neural approximation), the presented suboptimal algorithms do not require demanding on-line nonlinear optimization The presented simulation results demonstrate high accuracy and computational efficiency of the algorithms For a few representative nonlinear benchmark processes, such as chemical reactors and a distillation column, for which the classical MPC algorithms based on linear models do not work properly, the trajectories obtained in the suboptimal MPC algorithms are very similar to those given by the ``ideal'' MPC algorithm with on-line nonlinear optimization repeated at each sampling instant At the same time, the suboptimal MPC algorithms are significantly less computationally demanding

Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization

Abstract: The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.