Showing papers on "Convex optimization published in 2000"

Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

Abstract: In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimization problems, as well as generalizations of the structured singular value ($mu$) LMIs, have exactly this property. In the same spirit of exploiting structure to achieve computational efficiency, an algorithm for the numerical solution of a special class of frequency-dependent LMIs is presented. These optimization problems arise from robustness analysis questions, via the Kalman-Yakubovich-Popov lemma. The procedure is an outer approximation method based on the algorithms used in the computation of hinf norms for linear, time invariant systems. The result is especially useful for systems with large state dimension. The other main contribution in this thesis is the formulation of a convex optimization framework for semialgebraic problems, i.e., those that can be expressed by polynomial equalities and inequalities. The key element is the interaction of concepts in real algebraic geometry (Positivstellensatz) and semidefinite programming. To this end, an LMI formulation for the sums of squares decomposition for multivariable polynomials is presented. Based on this, it is shown how to construct sufficient Positivstellensatz-based convex tests to prove that certain sets are empty. Among other applications, this leads to a nonlinear extension of many LMI based results in uncertain linear system analysis. Within the same framework, we develop stronger criteria for matrix copositivity, and generalizations of the well-known standard semidefinite relaxations for quadratic programming. Some applications to new and previously studied problems are presented. A few examples are Lyapunov function computation, robust bifurcation analysis, structured singular values, etc. It is shown that the proposed methods allow for improved solutions for very diverse questions in continuous and combinatorial optimization.

A Modified Forward-Backward Splitting Method for Maximal Monotone Mappings

Abstract: We consider the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for monotone variational inequalities, under which the method converges assuming only the forward mapping is (Lipschitz) continuous on some closed convex subset of its domain. The modification entails an additional forward step and a projection step at each iteration. Applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed.

Piecewise linear quadratic optimal control

Abstract: The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy.

Control with random communication delays via a discrete-time jump system approach

Abstract: Digital control systems with random but bounded delays in the feedback loop can be modeled as finite-dimensional, discrete-time jump linear systems, with the transition jumps being modeled as finite-state Markov chains. This type of system can be called a "stochastic hybrid system". Due to the structure of the augmented state-space model, control of such a system is an output feedback problem, even if a state feedback law is intended for the original system. We present a V-K iteration algorithm to design switching and non-switching controllers for such systems. This algorithm uses an outer iteration loop to perturb the transition probability matrix. Inside this loop, one or more steps of V-K iteration is used to do controller synthesis, which requires the solution of two convex optimization problems constrained by LMIs.

Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

Abstract: Introduction. 1. Totally Convex Functions. 2. Computation of Fixed Points. 3. Infinite Dimensional Optimization. Bibliography. Index.

Finding Ultimate Limits of Performance for Hybrid Electric Vehicles

Abstract: Hybrid electric vehicles are seen as a solution to improving fuel economy and reducing pollution emissions from automobiles. By recovering kinetic energy during braking and optimizing the engine operation to reduce fuel consumption and emissions, a hybrid vehicle can outperform a traditional vehicle. In designing a hybrid vehicle, the task of finding optimal component sizes and an appropriate control strategy is key to achieving maximum fuel economy. In this paper we introduce the application of convex optimization to hybrid vehicle optimization. This technique allows analysis of the propulsion system’s capabilities independent of any specific control law. To illustrate this, we pose the problem of finding optimal engine operation in a pure series hybrid vehicle over a fixed drive cycle subject to a number of practical constraints including: • nonlinear fuel/power maps • min and max battery charge • battery efficiency • nonlinear vehicle dynamics and losses • drive train efficiency • engine slew rate limits We formulate the problem of optimizing fuel efficiency as a nonlinear convex optimization problem. This convex problem is then accurately approximated as a large linear program. As a result, we compute the globally minimum fuel consumption over the given drive cycle. This optimal solution is the lower limit of fuel consumption that any control law can achieve for the given drive cycle and vehicle. In fact, this result provides a means to evaluate a realizable control law's performance. We carry out a practical example using a spark ignition engine with lead acid (PbA) batteries. We close by discussing a number of extensions that can be done to improve the accuracy and versatility of these methods. Among these extensions are improvements in accuracy, optimization of emissions and extensions to other hybrid vehicle architectures. INTRODUCTION Two areas of significant importance in automotive engineering are improvement in fuel economy and reduction of emissions. Hybrid electric vehicles are seen as a means to accomplish these goals. The majority of vehicles in production today consist of an engine coupled to the road through a torque converter and a transmission with several fixed gear ratios. The transmission is controlled to select an optimal gear for the given load conditions. During braking, velocity is reduced by converting kinetic energy into heat. For the purposes of this introduction, it is convenient to consider two propulsion architectures: pure parallel and pure series hybrid vehicles. A parallel hybrid vehicle couples an engine to the road through a transmission. However, there is an electric motor that can be used to change the RPM and/or torque seen by the engine. In addition to modifying the RPM and/or torque, this motor can recover kinetic energy during braking and store it in a battery. By changing engine operating points and recovering kinetic energy, fuel economy and emissions can be improved. A series hybrid vehicle electrically couples the engine to the road. The propulsion system consists of an engine, a battery and an electric motor. The engine is a power source that is used to provide electrical power. The electrical power is used to recharge a battery or drive a motor. The motor propels the vehicle. This motor can also be used to recover kinetic energy during braking. For a given type of hybrid vehicle, there are three questions of central importance: • What are the important engine, battery and motor requirements? • When integrated into a vehicle, what is the best performance that can be achieved? • How closely does a control law approach this best performance? Answers to these questions can be found by solving three separate problems: • Solving for the maximum fuel economy that can be obtained for a fixed vehicle configuration on a fixed drive cycle independent of a control law. • Given a method to find maximum fuel economy, vary the vehicle component characteristics to find the optimal fuel economy. • Apply the selected control law to the system and determine the fuel consumption. Calculate the ratio between this control law’s fuel consumption and the optimal value to give a metric for how close the control law comes to operating the vehicle at its maximum performance. There are many hybrid vehicle architectures[1]. For the sake of simplicity, a pure series hybrid was chosen for this study. However, the methods used for series hybrid vehicles can be extended to apply to other hybrid vehicle architectures. This study was restricted to minimizing fuel economy. This method can be extended to include emissions. DISCUSSION: FINDING THE MAXIMUM FUEL ECONOMY FOR A GIVEN VEHICLE There are many approaches that can be used to determine the maximum fuel economy that can be obtained by a particular vehicle over a particular drive cycle. One common approach is to select a control law and then optimize that control law for the system. Other techniques search through control law architectures and control parameters simultaneously. Since these techniques select a control law before beginning the optimization, the minimum fuel economy found is always a function of the control law. This leaves open the question of whether selection of a better control law could have resulted in better fuel economy. The approach presented here finds the minimal fuel consumption of the vehicle independent of any control law. Because a control law is not part of the optimization, the fuel economy found is the best possible. It is noncausal in that it finds the minimum fuel consumption using knowledge of future power demands and past power demands. Therefore it represents a limit of performance of a causal control law. Furthermore, since the problem is formulated as a convex problem and then a linear program, the minimum fuel consumption calculated is guaranteed to be the global minimum solution. The discussion that follows details: 1. The formulation of the fuel economy minimization problem as a convex problem. 2. The reduction of this convex problem to a linear program. 3. Solution of the linear program to find the minimum fuel consumption. DESCRIBING THE PROBLEM To solve for maximum fuel economy, a model of the series hybrid vehicle is used. To simplify the model, the following assumptions are made: • The voltage on the electrical bus is constant. Voltage droop and ripple can be ignored. • The relationship between power output from the engine and fuel consumption can be assumed to be a fixed relationship that is not affected by transients. • The battery’s storage efficiency is constant. It does not change with state of charge or power levels. These simplifications are used to reduce the complexity of the resulting linear program and to maintain a problem description which is convex. These simplifications illustrate one of the challenges that arises in the application of convex analysis to engineering problems – finding a description of the problem which is convex.

Robust linear and support vector regression

Abstract: The robust Huber M-estimator, a differentiable cost function that is quadratic for small errors and linear otherwise, is modeled exactly, in the original primal space of the problem, by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators. Previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust Huber linear estimator. Numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems. Results are shown on problems with as many as 20000 data points, with considerably faster running times on larger problems.

A Survey of Algorithms for Convex Multicommodity Flow Problems

Abstract: Routing problems appear frequently when dealing with the operation of communication or transportation networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems. This paper emphasizes the message routing problem in data networks, but it includes a broader literature overview of convex multicommodity flow problems. We present and discuss the main solution techniques proposed for solving this class of large-scale convex optimization problems. We conduct some numerical experiments on the message routing problem with four different techniques.

H 2 and $H_\infty$ Robust Filtering for Discrete-Time Linear Systems

Abstract: This paper investigates robust filtering design problems in H2 and $H_\infty$ spaces for discrete-time systems subjected to parameter uncertainty which is assumed to belong to a convex bounded polyhedral domain. It is shown that, by a suitable change of variables, both design problems can be converted into convex programming problems written in terms of linear matrix inequalities (LMI). The results generalize the ones available in the literature to date in several directions. First, all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured. Then, assuming the order of the uncertain system is known, the optimal guaranteed performance H2 and $H_\infty$ filters are proven to be of the same order as the order of the system. Comparisons with robust filters for systems subjected to norm-bounded uncertainty are provided in both theoretical and practical settings. In particular, it is shown that under the same assumptions the results here are generally better as far as the minimization of a guaranteed cost expressed in terms of H2 or $H_\infty$ norms is considered. Some numerical examples illustrate the theoretical results.

Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization

Abstract: This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in \realn can be reformulated into certain smooth/convex unconstrained dual problems in \realm with m≤slant n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.

Extremal Problems and Isotropic Positions of Convex Bodies

Abstract: LetK be a convex body in ℝn and letWi(K),i=1, …,n−1 be its quermassintegrals. We study minimization problems of the form min{Wi(TK)|T ∈ SLn} and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropic characterization for appropriate measures. This shows that several well known positions of convex bodies which play an important role in the local theory may be described in terms of classical convexity as isotropic ones. We provide new applications of this point of view for the minimal mean width position.

Global Optimality Conditions for Quadratic Optimization Problems with Binary Constraints

Abstract: We consider nonconvex quadratic optimization problems with binary constraints. Our main result identifies a class of quadratic problems for which a given feasible point is global optimal. We also establish a necessary global optimality condition. These conditions are expressed in a simple way in terms of the problem's data. We also study the relations between optimal solutions of the nonconvex binary quadratic problem versus the associated relaxed and convex problem defined over the $l_{\infty}$ norm. Our approach uses elementary arguments based on convex duality.

Interior-point methods for nonconvex nonlinear programming: Orderings and higher-order methods

Abstract: The paper extends prior work by the authors on loqo, an interior point algorithm for nonconvex nonlinear programming. The specific topics covered include primal versus dual orderings and higher order methods, which attempt to use each factorization of the Hessian matrix more than once to improve computational efficiency. Results show that unlike linear and convex quadratic programming, higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming, but that a variant of Mehrotra’s predictor-corrector algorithm can definitely improve performance.

MA estimation in polynomial time

Abstract: The parameter estimation of moving-average (MA) signals from second-order statistics was deemed for a long time to be a difficult nonlinear problem for which no computationally convenient and reliable solution was possible. We show how the problem of MA parameter estimation from sample covariances can be formulated as a semidefinite program that can be solved in a time that is a polynomial function of the MA order. Two methods are proposed that rely on two specific (over) parametrizations of the MA covariance sequence, whose use makes the minimization of a covariance fitting criterion a convex problem. The MA estimation algorithms proposed here are computationally fast, statistically accurate, and reliable. None of the previously available algorithms for MA estimation (methods based on higher-order statistics included) shares all these desirable properties. Our methods can also be used to obtain the optimal least squares approximant of an invalid (estimated) MA spectrum (that takes on negative values at some frequencies), which was another long-standing problem in the signal processing literature awaiting a satisfactory solution.

Test-case generator for nonlinear continuous parameter optimization techniques

Abstract: The experimental results reported in many papers suggest that making an appropriate a priori choice of an evolutionary method for a nonlinear parameter optimization problem remains an open question. It seems that the most promising approach at this stage of research is experimental, involving the design of a scalable test suite of constrained optimization problems, in which many features could be tuned easily. It would then be possible to evaluate the merits and drawbacks of the available methods, as well as to test new methods efficiently. In this paper, we propose such a test-case generator for constrained parameter optimization techniques. This generator is capable of creating various test problems with different characteristics including: 1) problems with different relative sizes of the feasible region in the search space; 2) problems with different numbers and types of constraints; 3) problems with convex or nonconvex evaluation functions, possibly with multiple optima; and 4) problems with highly nonconvex constraints consisting of (possibly) disjoint regions. Such a test-case generator is very useful for analyzing and comparing different constraint-handling techniques.

Conic convex programming and self-dual embedding

Abstract: How ro initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the two-phase approach or using the so-called big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem. Viz. the self-dual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving tts embedded self-dual problem The embedded self-dual problem has a trivial initial solution and has the same stmctare as the original problem. Hence. it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programmng. Since a nonlinear conic convex programming problem may lack the so-called strict complementarity prop...

Text categorizers based on regularizing adaptations of the problem of computing linear separators

Abstract: A method to automatically categorize messages or documents containing text. The method of solution fits in the general framework of supervised learning, in which a rule or rules for categorizing data is automatically constructed by a computer on the basis of training data that has been labeled beforehand. More specifically, the method involves the construction of a linear separator: training data is used to construct for each category a weight vector w and a threshold t, and the decision of whether a hitherto unseen document d is in the category will depend on the outcome of the test w T x≧t, where x is a vector derived from the document d. The method also uses a set L of features selected from the training data in order to construct the numerical vector representation x of a document. The preferred method uses an algorithm based on Gauss-Seidel iteration to determine the weight factor w that is determined by a regularized convex optimization problem derived from the principle of minimizing modified training error.

Collapse behaviour of three-dimensional brick-block systems using non-linear programming

Abstract: A two-step procedure for the application of non linear constrained programming to the limit analysis of rigid brick-block systems with no-tension and frictional interface is implemented and applied to various masonry structures. In the first step, a linear problem of programming, obtained by applying the upper bound theorem of limit analysis to systems of blocks interacting through no-tension and dilatant interfaces, is solved. The solution of this linear program is then employed as initial guess for a non linear and non convex problem of programming, obtained applying both the \'mechanism\' and the \'equilibrium\' approaches to the same block system with no-tension and frictional interfaces. The optimiser used is based on the sequential quadratic programming. The gradients of the constraints required are provided directly in symbolic form. Ln this way the program easily converges to the optimal solution even for systems with many degrees of freedom. Various numerical analyses showed that the procedure allows a reliable investigation of the ultimate behaviour of jointed structures, such as stone masonry structures, under statical load conditions.

Neurocomputing with time delay analysis for solving convex quadratic programming problems

Abstract: This paper presents a neural-network computational scheme with time-delay consideration for solving convex quadratic programming problems. Based on some known results, a delay margin is explicitly determined for the stability of the neural dynamics, under which the states of the neural network does not oscillate. The configuration of the proposed neural network is provided. Operational characteristics of the neural network are demonstrated via numerical examples.

Robust non-fragile LQ controllers: The static state feedback case

Abstract: This paper describes the synthesis of non-fragile or resilient regulators for linear systems. A general framework for fragility is described using state-space methodologies, and the LQ/H2 static state-feedback problem is examined in detail. We discuss the multiplicative structured uncertainties case, and propose remedies of the fragility problem using an optimization programming framework via matrix inequalities. A special case that leads to a convex optimization framework via linear matrix inequalities (LMIs) will be considered. The benchmark problem is taken as an example to show how special controller gain variations can affect the performance of the closed-loop system.

Strong Convergence of Block-Iterative Outer Approximation Methods for Convex Optimization

Abstract: The strong convergence of a broad class of outer approximation methods for minimizing a convex function over the intersection of an arbitrary number of convex sets in a reflexive Banach space is studied in a unified framework. The generic outer approximation algorithm under investigation proceeds by successive minimizations over the intersection of convex supersets of the feasibility set determined in terms of the current iterate and variable blocks of constraints. The convergence analysis involves flexible constraint approximation and aggregation techniques as well as relatively mild assumptions on the constituents of the problem. Various well-known schemes are recovered as special realizations of the generic algorithm and parallel block-iterative extensions of these schemes are devised within the proposed framework. The case of inconsistent constraints is also considered.

Composite structures optimization using sequential convex programming

Abstract: The design of composite structures is considered here. The approximation concepts approach is used to solve the optimisation problem. The convex approximations of the MMA family are briefly described. Modifications of these approximations are presented. They are based on gradient information at two successive iterations, avoiding the use of the expensive second order derivatives. A two points fitting scheme is also described, where the function value at the preceding design point is used to improve the approximation. Numerical examples compare these schemes to the existing ones for the selection of optimal fibres orientations in laminates. It is shown how these two points based approximations are well adapted to the problem and can improve the optimisation task, leading to reasonable computational efforts. A procedure is also derived for considering simultaneously monotonous and non monotonous structural behaviours. The resulting generalised approximation scheme is well suited for the optimisation of composite structures when both plies thickness and fibres orientations are considered as design variables. All the studied approximations are convex and separable: the optimisation problem can then be solved using a dual approach.

Weak and strong convergence of solutions to accretive operator inclusions and applications

Abstract: Our purpose in this paper is to approximate solutions of accretive operators in Banach spaces. Motivated by Halpern's iteration and Mann's iteration, we prove weak and strong convergence theorems for resolvents of accretive operators. Using these results, we consider the convex minimization problem of finding a minimizer of a proper lower semicontinuous convex function and the variational problem of finding a solution of a variational inequality.

Stability analysis for neutral delay-differential systems

Abstract: In this paper, the problem of the stability analysis for neutral delay-differential systems is investigated. Using Lyapunov method, we present new sufficient conditions for the stability of the systems in terms of linear matrix inequality (LMI) which can be easily solved by various convex optimization algorithms. Numerical examples are given to illustrate the application of the proposed method.

Interior point methods : current status and future directions

Abstract: This article provides a synopsis of the major developments in interior point methods for mathematical programming in the last thirteen years, and discusses current and future research directions in interior point methods, with a brief selective guide to the research literature.1