Showing papers on "Nonlinear programming published in 2012"

CVXGEN: a code generator for embedded convex optimization

Abstract: CVXGEN is a software tool that takes a high level description of a convex optimization problem family, and automatically generates custom C code that compiles into a reliable, high speed solver for the problem family. The current implementation targets problem families that can be transformed, using disciplined convex programming techniques, to convex quadratic programs of modest size. CVXGEN generates simple, flat, library-free code suitable for embedding in real-time applications. The generated code is almost branch free, and so has highly predictable run-time behavior. The combination of regularization (both static and dynamic) and iterative refinement in the search direction computation yields reliable performance, even with poor quality data. In this paper we describe how CVXGEN is implemented, and give some results on the speed and reliability of the automatically generated solvers.

Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

Abstract: The growing costs of fuel and operation of power generating units warrant improvement of optimization methodologies for economic dispatch (ED) problems. The practical ED problems have non-convex objective functions with equality and inequality constraints that make it much harder to find the global optimum using any mathematical algorithms. Modern optimization algorithms are often meta-heuristic, and they are very promising in solving nonlinear programming problems. This paper presents a novel approach to determining the feasible optimal solution of the ED problems using the recently developed Firefly Algorithm (FA). Many nonlinear characteristics of power generators, and their operational constraints, such as generation limitations, prohibited operating zones, ramp rate limits, transmission loss, and nonlinear cost functions, were all contemplated for practical operation. To demonstrate the efficiency and applicability of the proposed method, we study four ED test systems having non-convex solution spaces and compared with some of the most recently published ED solution methods. The results of this study show that the proposed FA is able to find more economical loads than those determined by other methods. This algorithm is considered to be a promising alternative algorithm for solving the ED problems in practical power systems.

Minimum Loss Network Reconfiguration Using Mixed-Integer Convex Programming

Abstract: This paper proposes a mixed-integer conic programming formulation for the minimum loss distribution network reconfiguration problem. This formulation has two features: first, it employs a convex representation of the network model which is based on the conic quadratic format of the power flow equations and second, it optimizes the exact value of the network losses. The use of a convex model in terms of the continuous variables is particularly important because it ensures that an optimal solution obtained by a branch-and-cut algorithm for mixed-integer conic programming is global. In addition, good quality solutions with a relaxed optimality gap can be very efficiently obtained. A polyhedral approximation which is amenable to solution via more widely available mixed-integer linear programming software is also presented. Numerical results on practical test networks including distributed generation show that mixed-integer convex optimization is an effective tool for network reconfiguration.

Distribution System Planning With Incorporating DG Reactive Capability and System Uncertainties

Abstract: Distributed generation (DG) systems are considered an integral part in future distribution system planning. The active and reactive power injections from DG units, typically installed close to the load centers, are seen as a cost-effective solution for distribution system voltage support, energy saving, and reliability improvement. This paper proposes a novel distribution system expansion planning strategy encompassing renewable DG systems with schedulable and intermittent power generation patterns. The reactive capability limits of different renewable DG systems covering wind, solar photovoltaic, and biomass-based generation units are included in the planning model and the system uncertainties such as load demand, wind speed, and solar radiation are also accounted using probabilistic models. The problem of distribution system planning with renewable DG is formulated as constrained mixed integer nonlinear programming, wherein the total cost will be minimized with optimal allocation of various renewable DG systems. A solution algorithm integrating TRIBE particle swarm optimization (TRIBE PSO) and ordinal optimization (OO) is developed to effectively obtain optimal and near-optimal solutions for system planners. TRIBE PSO, OO, and the proposed algorithm are applied to a practical test system and results are compared and presented.

CasADi -- A symbolic package for automatic differentiation and optimal control

Abstract: We present CasADi, a free, open-source software tool for fast, yet efficient solution of nonlinear optimization problems in general and dynamic optimization problems in particular. To the developer of algorithms for numerical optimization and to the advanced user of such algorithms, it offers a level of abstraction which is notably lower, and hence more flexible, than that of algebraic modeling languages such as AMPL or GAMS, but higher than working with a conventional automatic differentiation (AD) tool.CasADi is best described as a minimalistic computer algebra system (CAS) implementing automatic differentiation in eight different flavors. Similar to algebraic modeling languages, it includes high-level interfaces to state-of-the-art numerical codes for nonlinear programming, quadratic programming and integration of differential-algebraic equations. CasADi is implemented in self-contained C++ code and contains full-featured front-ends to Python and Octave for rapid prototyping. In this paper, we present the AD framework of CasADi and benchmark the tool against AMPL for a set of nonlinear programming problems from the CUTEr test suite.

Coordination of Directional Overcurrent Relays Using Seeker Algorithm

Abstract: Coordination of directional overcurrent relays in a multiloop subtransmission or distribution network is formulated as an optimization problem. In this paper, the coordination of directional overcurrent relays is formulated as a mixed-integer nonlinear programming problem and is then solved by a new seeker optimization technique. Based on the act of human searching, in the proposed seeker technique, the search direction and step length are determined in an adaptive way. The proposed method is implemented in three different test cases. The results are compared with previously proposed analytic and evolutionary approaches.

Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms

Abstract: This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinate-wise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparse-simplex methods. The first algorithm is essentially a gradient projection method while the remaining two algorithms are of coordinate descent type. The theoretical convergence of these methods and their relations to the derived optimality conditions are studied. The algorithms and results are illustrated by several numerical examples.

Sequential Quadratic Programming Methods

Abstract: In his 1963 PhD thesis, Wilson proposed the first sequential quadratic programming (SQP) method for the solution of constrained nonlinear optimization problems. In the intervening 48 years, SQP methods have evolved into a powerful and effective class of methods for a wide range of optimization problems. We review some of the most prominent developments in SQP methods since 1963 and discuss the relationship of SQP methods to other popular methods, including augmented Lagrangian methods and interior methods. Given the scope and utility of nonlinear optimization, it is not surprising that SQP methods are still a subject of active research. Recent developments in methods for mixed integer nonlinear programming (MINLP) and the minimization of functions subject to differential equation constraints has led to a heightened interest in methods that may be “warm started” from a good approximate solution. We discuss the role of SQP methods in these contexts

Scenario-Based Multiobjective Volt/Var Control in Distribution Networks Including Renewable Energy Sources

Abstract: This paper proposes a stochastic multiobjective framework for daily volt/var control (VVC), including hydroturbine, fuel cell, wind turbine, and photovoltaic powerplants The multiple objectives of the VVC problem to be minimized are the electrical energy losses, voltage deviations, total electrical energy costs, and total emissions of renewable energy sources and grid For this purpose, the uncertainty related to hourly load, wind power, and solar irradiance forecasts are modeled in a scenario-based stochastic framework A roulette wheel mechanism based on the probability distribution functions of these random variables is considered to generate the scenarios Consequently, the stochastic multiobjective VVC (SMVVC) problem is converted to a series of equivalent deterministic scenarios Furthermore, an Evolutionary Algorithm using the Modified Teaching-Learning-Algorithm (MTLA) is proposed to solve the SMVVC in the form of a mixed-integer nonlinear programming problem In the proposed algorithm, a new mutation method is taken into account in order to enhance the global searching ability and mitigate the premature convergence to local minima Finally, two distribution test feeders are considered as case studies to demonstrate the effectiveness of the proposed SMVVC

Automatic robust convex programming

Abstract: This paper presents the robust optimization framework in the modelling language YALMIP, which carries out robust modelling and uncertainty elimination automatically and allows the user to concentrate on the high-level model. While introducing the software package, a brief summary of robust optimization is given, as well as some comments on modelling and tractability of complex convex uncertain optimization problems.

Multilevel Optimization: Algorithms and Applications

Abstract: In many decision processes there is an hierarchy of decision-makers and decisions are taken at different levels in this hierarchy. Multilevel programming focuses on the whole hierarchy structure. In terms of modeling, the constraint domain associated with a multilevel programming problem is implicitly determined by a series of optimization problems which must be solved in a predetermined sequence. The field of multilevel optimization has become a well-known and important research field. Hierarchical structures can be found in scientific disciplines such as environment, ecology, biology, chemical engineering, mechanics, classification theory, databases, network design, transportation, game theory and economics. Moreover, new applications are constantly being introduced. This has stimulated the development of new theory and efficient algorithms. This volume contains 16 chapters written by various leading researchers and presents a cohesive authoritative overview of developments and applications in their emerging field of optimization. Audience: Researchers whose work involves the application of mathematical programming and optimization to hierarchical structures.

Tuning of PID controller based on a multiobjective genetic algorithm applied to a robotic manipulator

Abstract: Highlights? Multiobjective optimization finds a set of solutions called non-dominated solutions. ? The NSGA-II approach is evaluated. ? This algorithm is tested in PID tuning using a robotic manipulator of two-degree-of-freedom. Most controllers optimization and design problems are multiobjective in nature, since they normally have several (possibly conflicting) objectives that must be satisfied at the same time. Instead of aiming at finding a single solution, the multiobjective optimization methods try to produce a set of good trade-off solutions from which the decision maker may select one. Several methods have been devised for solving multiobjective optimization problems in control systems field. Traditionally, classical optimization algorithms based on nonlinear programming or optimal control theories are applied to obtain the solution of such problems. The presence of multiple objectives in a problem usually gives rise to a set of optimal solutions, largely known as Pareto-optimal solutions. Recently, Multiobjective Evolutionary Algorithms (MOEAs) have been applied to control systems problems. Compared with mathematical programming, MOEAs are very suitable to solve multiobjective optimization problems, because they deal simultaneously with a set of solutions and find a number of Pareto optimal solutions in a single run of algorithm. Starting from a set of initial solutions, MOEAs use iteratively improving optimization techniques to find the optimal solutions. In every iterative progress, MOEAs favor population-based Pareto dominance as a measure of fitness. In the MOEAs context, the Non-dominated Sorting Genetic Algorithm (NSGA-II) has been successfully applied to solving many multiobjective problems. This paper presents the design and the tuning of two PID (Proportional-Integral-Derivative) controllers through the NSGA-II approach. Simulation numerical results of multivariable PID control and convergence of the NSGA-II is presented and discussed with application in a robotic manipulator of two-degree-of-freedom. The proposed optimization method based on NSGA-II offers an effective way to implement simple but robust solutions providing a good reference tracking performance in closed loop.

Maximum Block Improvement and Polynomial Optimization

Abstract: In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we accept only a block update that achieves the maximum improvement, hence the name of our new search method: maximum block improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proved. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem; thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.

Computer-Aided Design and Optimization of High-Efficiency LLC Series Resonant Converter

Abstract: High conversion efficiency is desired in switch mode power supply converters. Computer-aided design optimization is emerging as a promising way to design power converters. In this work a systematic optimization procedure is proposed to optimize LLC series resonant converter full load efficiency. A mode solver technique is proposed to handle LLC converter steady-state solutions. The mode solver utilizes numerical nonlinear programming techniques to solve LLC-state equations and determine operation mode. Loss models are provided to calculate total component losses using the current and voltage information derived from the mode solver. The calculated efficiency serves as the objective function to optimize the converter efficiency. A prototype 300-W 400-V to 12-V LLC converter is built using the optimization results. Details of design variables, boundaries, equality/inequality constraints, and loss distributions are given. An experimental full-load efficiency of 97.07% is achieved compared to a calculated 97.4% efficiency. The proposed optimization procedure is an effective way to design high-efficiency LLC converters.

Unconstrained Optimization of Real Functions in Complex Variables

Abstract: Nonlinear optimization problems in complex variables are frequently encountered in applied mathematics and engineering applications such as control theory, signal processing, and electrical engineering. Optimization of these problems often requires a first- or second-order approximation of the objective function to generate a new step or descent direction. However, such methods cannot be applied to real functions of complex variables because they are necessarily nonanalytic in their argument, i.e., the Taylor series expansion in their argument alone does not exist. To overcome this problem, the objective function is usually redefined as a function of the real and imaginary parts of its complex argument so that standard optimization methods can be applied. However, this approach may needlessly disguise any inherent structure present in the derivatives of such complex problems. Although little known, it is possible to construct an expansion of the objective function in its original complex variables by noti...

Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

Abstract: In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.

On the optimal design of water distribution networks: a practical MINLP approach

Abstract: We propose a practical solution method for real-world instances of a water-network optimization problem with fixed topology using a nonconvex continuous NLP (NonLinear Programming) relaxation and a MINLP (Mixed Integer NonLinear Programming) search. Our approach employs a relatively simple and accurate model that pays some attention to the requirements of the solvers that we employ. Our view is that in doing so, with the goal of calculating only good feasible solutions, complicated algorithmics can be confined to the MINLP solver. We report successful computational experience using available open-source MINLP software on problems from the literature and on difficult real-world instances. An important contribution of this paper is that the solutions obtained, besides being low cost, are immediately usable in practice because they are characterized by an allocation of diameters to pipes that leads to a correct hydraulic operation of the network. This is not the case for most of the other methods presented in the literature.

Fuzzy AHP-based multicriteria decision making systems using particle swarm optimization

Abstract: This paper presents a fuzzy optimization model to solve multicriteria decision making (MCDM) systems based on a fuzzy analytic hierarchy process (fuzzy AHP). To deal with the imprecise judgments of decision makers, a fuzzy AHP decision making model is proposed as an evaluation tool, where the expert's comparison judgments are translated into fuzzy numbers. Unlike the conventional fuzzy AHP methods, the proposed method drives exact weights from consistent and inconsistent fuzzy comparison matrices, which eliminate the need of additional aggregation and ranking procedures. The proposed method transforms a fuzzy prioritization problem into a constrained nonlinear optimization model. An improved particle swarm optimization (PSO) is applied to solve the optimization model as a nonlinear system of equations. Several illustrative examples using existing fuzzy AHP methods are given to demonstrate the effectiveness of the proposed method.

Levelset based fluid topology optimization using the extended finite element method

Abstract: This study focuses on finding the optimal layout of fluidic devices subjected to incompressible flow at low Reynolds numbers. The proposed approach uses a levelset method to describe the fluid-solid interface geometry. The flow field is modeled by the incompressible Navier–Stokes equations and discretized by the extended finite element method (XFEM). The no-slip condition along the fluid-solid interface is enforced via a stabilized Lagrange multiplier method. Unlike the commonly used porosity approach, the XFEM approach does not rely on a material interpolation scheme, which allows for more flexibility in formulating the design problems. Further, it mitigates shortcomings of the porosity approach, including spurious pressure diffusion through solid material, strong dependency of the accuracy of the boundary enforcement with respect to the model parameters which may affect the optimization results, and poor boundary resolution. Numerical studies verify that the proposed method is able to recover optimization results obtained with the porosity approach. Further, it is demonstrated that the XFEM approach yields physical results for problems that cannot be solved with the porosity approach.

A modified particle swarm optimization for disaster relief logistics under uncertain environment

Abstract: Relief logistics is one of the most important elements of a relief operation. This paper investigates a relief chain design problem where not only demands but also supplies and the cost of procurement and transportation are considered as the uncertain parameters. Furthermore, the model considers uncertainty for the locations where those demands can arise and the possibility that a number of the facility could be partially destroyed by the disaster. The proposed model for this study is formulated as a mixed-integer nonlinear programming to minimize the sum of the expected total cost (which includes costs of location, procurement, transportation, holding, and shortage) and the variance of the total cost. The model simultaneously determines the location of relief distribution centers and the allocation of affected area to relief distribution centers. Furthermore, an efficient solution approach based on particle swarm optimization is developed in order to solve the proposed mathematical model. At last, computational results for several instances of the problem are presented to demonstrate the feasibility and effectiveness of the proposed model and algorithm.

The ESA NLP Solver WORHP

Abstract: We Optimize Really Huge Problems (WORHP) is a solver for large-scale, sparse, nonlinear optimization problems with millions of variables and constraints. Convexity is not required, but some smoothness and regularity assumptions are necessary for the underlying theory and the algorithms based on it. WORHP has been designed from its core foundations as a sparse sequential quadratic programming (SQP) / interior-point (IP) method; it includes efficient routines for computing sparse derivatives by applying graph-coloring methods to finite differences, structure-preserving sparse named after Broyden, Fletcher, Goldfarb and Shanno (BFGS) update techniques for Hessian approximations, and sparse linear algebra. Furthermore it is based on reverse communication, which offers an unprecedented level of interaction between user and nonlinear programming (NLP) solver. It was chosen by ESA as the European NLP solver on the basis of its high robustness and its application-driven design and development philosophy. Two large-scale optimization problems from space applications that demonstrate the robustness of the solver complement the cursory description of general NLP methods and some WORHP implementation details.

Fast and Global Optimal Energy-Efficient Control Allocation With Applications to Over-Actuated Electric Ground Vehicles

Abstract: This paper presents a fast and global optimization algorithm for an energy-efficient control allocation (CA) scheme, which was proposed for improving the operational energy efficiency of over-actuated systems. For a class of realistic actuator power and efficiency functions, a Karush-Kuhn-Tucker (KKT)-based algorithm was devised to find all the local optimal solutions, and consequently the global minimum through a further simple comparison among all the realistic local minima and boundary values for such a non-convex optimization problem. This KKT-based algorithm is also independent on the selections of initial conditions by transferring the standard nonlinear optimization problem into classical eigenvalue problems. Numerical examples for electric vehicles with in-wheel motors were utilized to validate the effectiveness of the proposed global optimization algorithm. Simulation results, based on the parameters of an electric ground vehicle actuated by in-wheel motors (whose energy efficiencies were experimentally calibrated), showed that the proposed global optimization algorithm was at least 20 times faster than the classical active-set optimization method, while achieving better control allocation results for system energy saving.

SIAM Journal on Optimization

Abstract: The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.

Multiobjective Optimal Location of FACTS Shunt-Series Controllers for Power System Operation Planning

Abstract: This paper develops appropriate models of flexible ac transmission systems (FACTS) shunt-series controllers for multiobjective optimization and also presents a multiobjective optimization methodology to find the optimal location of FACTS shunt-series controllers. The objective functions are the total fuel cost, power losses, and system loadability with and without minimum cost of FACTS installation. The e-constraint approach is implemented for the multiobjective mathematical programming (MMP) formulation, including the FACTS shunt-series controllers (i.e., phase-shifting transformer (PST), hybrid flow controller (HFC), and unified power-flow controller (UPFC)). Simulation results are presented for the IEEE 14-bus system. The optimization method is numerically solved using Matlab and general algebraic modeling system (GAMS) software environments. The solution procedure uses nonlinear programming (NLP) and mixed-integer nonlinear programming (MINLP) to solve the optimal location and setting of FACTS incorporated in the optimal power-flow problem considering these objective functions and improving the power system operation. Furthermore, the results demonstrate that the HFC is outperformed by PST and UPFC from the analytical and technical point of views.

Second-Order Subdifferential Calculus With Applications to Tilt Stability in Optimization

Abstract: This paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finite- dimensional spaces. The main focus is the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal second-order chain rule for strongly and fully amenable compositions. We also calculate the second- order subdifferentials for some major classes of piecewise linear-quadratic functions. These results are applied to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms.

Trading regret for efficiency: online convex optimization with long term constraints

Abstract: In this paper we propose efficient algorithms for solving constrained online convex optimization problems. Our motivation stems from the observation that most algorithms proposed for online convex optimization require a projection onto the convex set Κ from which the decisions are made. While the projection is straightforward for simple shapes (e.g., Euclidean ball), for arbitrary complex sets it is the main computational challenge and may be inefficient in practice. In this paper, we consider an alternative online convex optimization problem. Instead of requiring that decisions belong to Κ for all rounds, we only require that the constraints, which define the set Κ, be satisfied in the long run. By turning the problem into an online convex-concave optimization problem, we propose an efficient algorithm which achieves O(√T) regret bound and O(T3/4) bound on the violation of constraints. Then, we modify the algorithm in order to guarantee that the constraints are satisfied in the long run. This gain is achieved at the price of getting O(T3/4) regret bound. Our second algorithm is based on the mirror prox method (Nemirovski, 2005) to solve variational inequalities which achieves O(T2/3) bound for both regret and the violation of constraints when the domain Κ can be described by a finite number of linear constraints. Finally, we extend the results to the setting where we only have partial access to the convex set Κ and propose a multipoint bandit feedback algorithm with the same bounds in expectation as our first algorithm.

Optimal sensitivity based on IPOPT

Abstract: We introduce a flexible, open source implementation that provides the optimal sensitivity of solutions of nonlinear programming (NLP) problems, and is adapted to a fast solver based on a barrier NLP method. The program, called sIPOPT evaluates the sensitivity of the Karush–Kuhn–Tucker (KKT) system with respect to perturbation parameters. It is paired with the open-source IPOPT NLP solver and reuses matrix factorizations from the solver, so that sensitivities to parameters are determined with minimal computational cost. Aside from estimating sensitivities for parametric NLPs, the program provides approximate NLP solutions for nonlinear model predictive control and state estimation. These are enabled by pre-factored KKT matrices and a fix-relax strategy based on Schur complements. In addition, reduced Hessians are obtained at minimal cost and these are particularly effective to approximate covariance matrices in parameter and state estimation problems. The sIPOPT program is demonstrated on four case studies to illustrate all of these features.

Modeling and Optimization of Energy Consumption in Cooperative Multi-Robot Systems

Abstract: Reduction of energy consumption is important for reaching a sustainable future. This paper presents a novel method for optimizing the energy consumption of robotic manufacturing systems. The method embeds detailed evaluations of robots' energy consumptions into a scheduling model of the overall system. The energy consumption for each operation is modeled and parameterized as function of the operation execution time, and the energy-optimal schedule is derived by solving a mixed-integer nonlinear programming problem. The objective function for the optimization problem is then the total energy consumption for the overall system. A case study of a sample robotic manufacturing system and an experiment on an industrial robot are presented. They show that there exists a real possibility for a significant reduction of the energy consumption in comparison to state-of-the-art scheduling approaches.