Efficient Interior-point Algorithms Research Articles

Elastoplastic and limit analysis of 3D steel assemblies using second-order cone programming and dual finite-elements

We investigate the use of a second-order cone programming (SOCP) framework for computing complex 3D steel assemblies in the context of elastoplasticity and limit analysis. Displacement and stress-based variational formulations are considered and appropriate finite-element discretization strategies are chosen, yielding respectively an upper and lower bound estimate of the exact solution. An efficient interior-point algorithm is used to solve the associated optimization problems. The discrete solution convergence is estimated by comparing both static and kinematic solutions, offering a way to perform local mesh adaptation. The proposed framework is illustrated on the design of a moment-transmitting assembly, its performance is assessed by comparison with classical elastoplastic computations using Abaqus and, finally, T-stub resistance and failure mechanisms when assessing the strength of a column base plate are compared with the Eurocodes design rules.

The equal tendency algorithm: a new heuristic for the reliability model

The paper presents the equal tendency algorithm for approximating the reliability problem. The algorithm is based on the fact that a reliability product of various components tends to be maximal if the individual components are approximately equal. The nonlinear integer reliability problem is approximated as a binary linear integer model using the equal tendency theorem. The resulting linear model is then solved using the available efficient interior point algorithms.

Duality and profit efficiency for the hyperbolic measure model

The hyperbolic measure (HM) model is a radial, non-oriented model that is often used in Data Envelopment Analysis (DEA). It is formulated as a non-linear programming problem and hence the conventional linear programming methods, customarily used in DEA, cannot be applied to it in general. In this paper, we reformulate the hyperbolic measure model in a semidefinite programming framework which opens the way to solving the HM model by reliable and efficient interior point algorithms and allows us to benefit from simple primal-dual correspondence in semidefinite programming. We derive the dual of the HM model and so, for the first time, establish its multiplier form. We also offer an economic interpretation of the dual HM model via a comparison with the multiplier form of the directional distance model and relate the HM score to the so-called Nerlovian profit efficiency.

The solution of euclidean norm trust region SQP subproblems via second-order cone programs: an overview and elementary introduction

It is well known that convex sequential quadratic programming (SQP) subproblems with an Euclidean norm trust region constraint can be reduced to second-order cone programs for which the theory of Euclidean Jordan algebras leads to efficient interior-point algorithms. Here, a brief and self-contained outline of the principles of such an implementation is given and the application to SQP subproblems as well as to cubic regularization problems is discussed. All identities relevant for the implementation are derived from scratch and are compared to interior-point methods for linear programs (LPs). Sparsity of the data of the SQP subproblem can be maintained essentially in the same way as for interior-point methods for LPs. The presentation is intended as an introduction for students and for colleagues who may have heard about Jordan algebras but did not yet find the time to get involved with them. A simple Matlab implementation is made available and the discussion of implementational aspects addresses a scaling property that is critical for SQP subproblems.

An efficient interior-point algorithm with new non-monotone line search filter method for nonlinear constrained programming

ABSTRACTAn efficient primal-dual interior-point algorithm using a new non-monotone line search filter method is presented for nonlinear constrained programming, which is widely applied in engineering optimization. The new non-monotone line search technique is introduced to lead to relaxed step acceptance conditions and improved convergence performance. It can also avoid the choice of the upper bound on the memory, which brings obvious disadvantages to traditional techniques. Under mild assumptions, the global convergence of the new non-monotone line search filter method is analysed, and fast local convergence is ensured by second order corrections. The proposed algorithm is applied to the classical alkylation process optimization problem and the results illustrate its effectiveness. Some comprehensive comparisons to existing methods are also presented.

Plastic Collapse Analysis of Mindlin–Reissner Plates Using a Stabilized Mesh-Free Method

This paper presents a numerical kinematic formulation for computation of collapse load of Mindlin–Reissner plates, that uses a stabilized mesh-free method in combination with second-order cone programming (SOCP). The kinematic formulation is discretized using a moving least squares approximation combined with a stabilized conforming nodal integration scheme, ensuring that shear locking problem at thin plate limit can be removed. The stabilized mesh-free based kinematic formulation is formulated as a conic problem so that it can be solved by an efficient primal-dual interior-point algorithm. To speed up computational progress, an adaptive refinement scheme using dissipation-based error indicator is also performed. The performance of the proposed numerical procedure is illustrated by examining several plates with arbitrary geometries and various boundary conditions.

Energy optimized orthonormal wavelet filter bank with prescribed sharpness

Optimization with respect to some energy measure such as compaction energy is a widely used criterion for designing wavelet filter banks. The filter bank can be adapted to the signal that it is analyzing to achieve good performance. The frequency selectivity property of a traditional low-pass filter is however not ensured using this criterion. Frequency selectivity is important to ensure the effects on aliasing is minimized in the subband and to give a regular equivalent wavelet function. In this work the design of energy optimized filters with a prescribed sharpness in the frequency response is presented. The sharpness, which determines the degree of selectivity, is achieved by the zero-pinning technique on the Bernstein polynomial. The design technique can be cast as a Semidefinite Programming (SDP) problem which can be solved with efficient interior point algorithms.

Extended parameter-dependent H<inf>∞</inf> filtering for uncertain continuous-time state-delayed systems

The design of robust H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> filtering problem of polytopic uncertain linear time-delay systems is addressed. The uncertain parameters are supposed to reside in a polytope. A parameter-dependent Lyapunov function approach is proposed for the design of filters that ensure a prescribed H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> performance level for all admissible uncertain parameters, which is different from the quadratic framework that entails fixed matrices for the entire uncertainty domain. This idea is realized by carefully selecting the structure of the matrices involved in the products with system matrices. An extended H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> sufficient condition for the existence of robust estimators is formulated in terms of linear matrix inequalities, which can be solved via efficient interior-point algorithms.

Topology optimization in electromagnetic casting via quadratic programming

A new optimization method is proposed for solving an inverse problem concerning the shape and topology of the inductors used in the electromagnetic casting technique of the metallurgical industry. The method is based on an sparse convex quadratic programming version of a recently proposed topology optimization formulation of the inverse electromagnetic casting problem. Regular 0–1 solutions are found by adding to the original Kohn–Vogelius objective function an appropriate penalty term that preserves the quadratic programming structure of the problem, allowing the use of efficient interior-point algorithms. Results for some numerical examples are presented, showing that the technique proposed is effective and can successfully find inductors of optimal shape and topology.

A Fuzzy Lyapunov LMI Criterion to a Chaotic System

Abstract In terms of the Fuzzy Lyapunov method, this work proposes stability conditions for fuzzy logic control (FLC). Their application for chaotic systems can be approximated by the Tagagi-Sugeno (T-S) fuzzy model. The fuzzy Lyapunov function is defined as a fuzzy blending of quadratic Lyapunov functions. External forces or disturbances are not considered in the controlled systems. In the design controller procedure, a parallel distributed compensation (PDC) scheme is utilized to construct a global FLC by blending all linear local state feedback controllers. The stability criteria are found not only for fuzzy modeling but also for a real chaotic system. Furthermore, this controller design problem can be reduced to a linear matrix inequality (LMI) problem by use of the Schur Complements. Efficient interior-point algorithms are now available in Matlab toolbox to solve this type of problem.

ℋ ∞ filtering for linear continuous-time systems subject to sensor non-linearities

In this study, the H ∞ filtering problem for linear continuous-time systems subject to sensor non-linearities is considered. First, the global H ∞ filtering is addressed by modelling the sensor non-linearities as sector non-linearities. The existence condition for an H ∞ filter is derived for such systems by using both the circle criterion theory and Popov criterion theory. In both cases, design methods for H ∞ filtering are proposed by means of linear matrix inequalities (LMIs)-based optimisation approach, which enables solutions to be found via efficient interior-point algorithms. To reduce the conservatism in the design of global H ∞ filters for linear systems with output saturation non-linearities, the local H ∞ filtering problem is considered. Local H ∞ filtering refers that the states are within some bounded set while a regional L 2 gain is guaranteed. An LMI-based approach is developed to calculate parameter matrices of the local H ∞ filter. Numerical examples are given to demonstrate the usefulness of the proposed results.

A preliminary set of applications leading to stochastic semidefinite programs and chance-constrained semidefinite programs

Ariyawansa and Zhu have recently introduced (two-stage) stochastic semidefinite programs (with recourse) (SSDPs) [1] and chance-constrained semidefinite programs (CCSDPs) [2] as paradigms for dealing with uncertainty in applications leading to semidefinite programs. Semidefinite programs have been the subject of intense research during the past 15 years, and one of the reasons for this research activity is the novelty and variety of applications of semidefinite programs. This research activity has produced, among other things, efficient interior point algorithms for semidefinite programs. Semidefinite programs however are defined using deterministic data while uncertainty is naturally present in applications. The definitions of SSDPs and CCSDPs in [1,2] were formulated with the expectation that they would enhance optimization modeling in applications that lead to semidefinite programs by providing ways to handle uncertainty in data. In this paper, we present results of our attempts to create SSDP and CCSDP models in four such applications. Our results are promising and we hope that the applications presented in this paper would encourage researchers to consider SSDP and CCSDP as new paradigms for stochastic optimization when they formulate optimization models.

Optimal Siting and Sizing of UPFC Control Settings in Grid Integrated Wind Energy Conversion Systems

The depletion of fossil fuel reserves, emission of greenhouse gases and the uneven distribution of existing reserves led the countries to look for sustainable alternatives, especially wind power. In India mostly Squirrel cage induction generators (SCIG) are used for extracting energy from the wind. Induction generators inject real power to the grid and absorb reactive power from grid. Normally, fixed capacitors of rating equal to no-load compensation are installed at the wind-turbine. Reactive power absorbed by the SCIG over and above the no-lad compensation is dependent on the operating condition. To compensate for the reactive power (over and above the no-load compensation) dynamic VAR compensator can be installed at the point of common coupling. When the wind-farm is connected to a weak grid there may be a problem with wind penetration into the grid. UPFC (a versatile FACTS controller) will be able to alleviate the problems associated with fixed speed wind-farms that are connected to a weak grid. In this paper finding the location and capacity of the UPFC for minimisation of power generation cost is posed as a non-linear optimization problem. An efficient Primal-Dual Interior Point algorithm in conjunction with second order sensitivity analysis is made use for solving the above problem. The optimal line placement for wind penetration in terms of marginal values of UPFC variables are identified using first order sensitivity analysis. Second order sensitivity analysis has been employed to identify the optimal line placement for highest cost savings. Further actual cost savings and optimal control settings of UPFC are evaluated by actually placing UPFC in each line. The proposed approach is tested on a sample 9-bus system using the program developed in Matlab and the results are encouraging. The results indicate that the estimation of optimal placement of UPFC for a large system is possible reducing the computation time involved.

Improved H ∞ filtering for Markov jumping linear systems with non-accessible mode information

This paper is concerned with the H∞ filtering problems for both continuous- and discrete-time Markov jumping linear systems (MJLS) with non-accessible mode information. A new design method is proposed, which greatly reduces the overdesign introduced in the derivation process. The desired filters can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities (LMIs), which can be solved via efficient interior-point algorithms. Numerical examples are provided to illustrate the advantages of the proposed approach.

Minimum Vibration Mechanism Design Via Convex Programming

One means of designing reduced vibration mechanisms is to ensure that the mechanism’s natural frequency be sufficiently greater than the driving frequencies of the actuators. In this paper we consider the problem of determining a mechanism’s mass, inertial, and joint stiffness parameters so as to maximize the lowest natural frequency of the mechanism. We show that this leads to a convex programming problem, which is characterized by a global optima that can be found with efficient interior point algorithms. Several case studies involving open and closed chain mechanisms demonstrate the viability of the design methodology.

Parameter-dependent robust [formula omitted] filtering for uncertain discrete-time systems

This paper is concerned with the problem of parameter-dependent H ∞ filtering for discrete-time systems with polytopic uncertainties. The uncertain parameters are supposed to reside in a polytope. Being different from previous results in the quadratic framework, the parameter-dependent Lyapunov function is used in this paper. Both full- and reduced-order filters are designed, which guarantee the asymptotic stability and a prescribed H ∞ performance level. The filter parameters can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities, which can be solved via efficient interior-point algorithms. Numerical examples are presented to illustrate the feasibility and less conservativeness of the proposed method.

Convex optimization of error probability of multi-antenna broadcast channel

In this paper, we propose a nonlinear convex optimization problem to minimize the average probability of error of all the users, subject to the peak or the average power constraints in the multi-input–multi-output broadcast channel. The proposed transmitter represents a nonlinear one-to-one mapping between the transmitted data vector and the symbol vector. The transmitted data vector going through the base station antennas is obtained as a solution to the proposed convex error probability optimization problem that can be solved using computationally efficient interior point algorithms. It is proved that the proposed transmitter is always optimal unless the signal-to-noise ratio (SNR) is too low or the channel is nearly singular. Simulation results demonstrate that the proposed transmission scheme significantly improves the probability of error as compared to several earlier approaches to multi-antenna broadcast transmitter design.

Dynamic Optimal Power Flow Using Interior Point Method and Benders Decomposition Considering Active and Reactive Constraints

This article describes an approach for the dynamic optimal power flow problem using interior point method (IPM) and Benders decomposition considering both active and reactive constraints. An efficient predictor-corrector primal-dual interior-point algorithm is used to solve the linearized dynamic OPF problem. The inclusion of active and reactive security constraints will assure a subsequent feasible solution for the dynamic OPF problem. The dynamic OPF problem is decomposed into a master problem and subproblems for checking the feasibility of the active and reactive constraints. The master problem is formulated and solved without active and reactive constraints to avoid complexity. The obtained dynamic OPF schedule from the master problem is applied to the active and reactive subproblems to minimize the violations. In case the current mix of scheduled units cannot remove the violations, additional constraints will be introduced in the master problem for rescheduling dynamic OPF. The proposed approach has been evaluated on an IEEE 118-bus test system. The results obtained with the proposed approach are presented and compared favorably with results of other stochastic techniques.

An Efficient Interior-Point Method for Convex Multicriteria Optimization Problems

In multicriteria optimization, several objective functions have to be minimized simultaneously. We propose a new efficient method for approximating the solution set of a multicriteria optimization problem, where the objective functions involved are arbitrary convex functions and the set of feasible points is convex. The method is based on generating warm-start points for an efficient interior-point algorithm, while the approximation computed consists of a finite set of discrete points. Polynomial-time complexity results for the method proposed are derived. In these estimates, the number of operations per point decreases when the number of points generated for the approximation increases. This reduced theoretical complexity estimate is a novel feature and is not observed in standard solution techniques for multicriteria optimization problems.

Stability conditions of fuzzy systems and its application to structural and mechanical systems

This paper provided the stability conditions and controller design for a class of structural and mechanical systems represented by Takagi–Sugeno (T–S) fuzzy models. In the design procedure of controller, parallel-distributed compensation (PDC) scheme was utilized to construct a global fuzzy logic controller by blending all local state feedback controllers. A stability analysis was carried out not only for the fuzzy model but also for a real mechanical system. Furthermore, this control problem can be reduced to linear matrix inequalities (LMI) problems by the Schur complements and efficient interior-point algorithms are now available in Matlab toolbox to solve this problem. A simulation example was given to show the feasibility of the proposed fuzzy controller design method.