Showing papers on "Convex optimization published in 1996"

Semidefinite programming

Abstract: In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.

An approach to fuzzy control of nonlinear systems: stability and design issues

Abstract: Presents a design methodology for stabilization of a class of nonlinear systems. First, the authors represent a nonlinear plant with a Takagi-Sugeno fuzzy model. Then a model-based fuzzy controller design utilizing the concept of the so-called "parallel distributed compensation" is employed. The main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system. The design procedure is conceptually simple and natural. Moreover, the stability analysis and control design problems can be reduced to linear matrix inequality (LMI) problems. Therefore, they can be solved efficiently in practice by convex programming techniques for LMIs. The design methodology is illustrated by application to the problem of balancing and swing-up of an inverted pendulum on a cart.

On Projection Algorithms for Solving Convex Feasibility Problems

Abstract: Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.

Linear Programming: Foundations and Extensions

Abstract: Preface. Part 1: Basic Theory - The Simplex Method and Duality. 1. Introduction. 2. The Simplex Method. 3. Degeneracy. 4. Efficiency of the Simplex Method. 5. Duality Theory. 6. The Simplex Method in Matrix Notation. 7. Sensitivity and Parametric Analyses. 8. Implementation Issues. 9. Problems in General Form. 10. Convex Analysis. 11. Game Theory. 12. Regression. Part 2: Network-Type Problems. 13. Network Flow Problems. 14. Applications. 15. Structural Optimization. Part 3: Interior-Point Methods. 16. The Central Path. 17. A Path-Following Method. 18. The KKT System. 19. Implementation Issues. 20. The Affine-Scaling Method. 21. The Homogeneous Self-Dual Method. Part 4: Extensions. 22. Integer Programming. 23. Quadratic Programming. 24. Convex Programming. Appendix A: Source Listings. Answers to Selected Exercises. Bibliography. Index.

Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems

Abstract: This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.

Induced L2-norm control for LPV systems with bounded parameter variation rates

Abstract: A linear, finite-dimensional plant, with state-space parameter dependence, is controlled using a parameter-dependent controller. The parameters whose values are in a compact set, are known in real time. Their rates of variation are bounded and known in real time also. The goal of control is to stabilize the parameter-dependent closed-loop system, and provide disturbance/error attenuation as measured in induced L2 norm. Our approach uses a bounding technique based on a parameter-dependent Lyapunov function, and then solves the control synthesis problem by reformulating the existence conditions into a semi-infinite dimensional convex optimization. We propose finite dimensional approximations to get sufficient conditions for successful controller design.

A new algorithm for minimizing convex functions over convex sets

Abstract: Let $$S \subseteq \mathbb{R}^n $$ be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ n the oracle returns a “Yes” ifz∈S; whereas ifz∉S then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.

Barrier functions in interior point methods

Abstract: We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for homogeneous cones. We identify some known barriers as the universal barrier scaled by an appropriate constant. We also calculate some new universal barrier functions. Our results connect the field of interior point methods to several branches of mathematics such as Lie groups, Jordan algebras, Siegel domains, differential geometry, complex analysis of several variables, etc.

Hidden convexity in some nonconvex quadratically constrained quadratic programming

Abstract: We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints. Under a suitable simultaneous diagonalization assumption (which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints, which is also shown to be equivalent to a minimax convex problem. In both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts. Special cases and applications are also discussed. We outline interior-point polynomial-time algorithms for the solution of the equivalent convex programs.

LMI optimization for nonstandard Riccati equations arising in stochastic control

Abstract: We consider coupled Riccati equations that arise in the optimal control of jump linear systems. We show how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs). The results extend to other nonstandard Riccati equations that arise, e.g., in the optimal control of linear systems subject to state-dependent multiplicative noise. Some nonstandard Riccati equations (such as those connected to linear systems subject to both state- and control-dependent multiplicative noise) are not amenable to the method. We show that we can still use LMI optimization to compute the optimal control law for the underlying control problem without solving the Riccati equation.

Occupation measures for controlled Markov processes: characterization and optimality

Abstract: For controlled Markov processes taking values in a Polish space, control problems with ergodic cost, infinite-horizon discounted cost and finite-horizon cost are studied. Each is posed as a convex optimization problem wherein one tries to minimize a linear functional on a closed convex set of appropriately defined occupation measures for the problem. These are characterized as solutions of a linear equation asssociated with the problem. This characterization is used to establish the existence of optimal Markov controls. The dual convex optimization problem is also studied.

Convex analysis of output feedback control problems: robust stability and performance

Abstract: This paper addresses the problem of optimal H/sub 2/ control by output feedback. Necessary and sufficient conditions on the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function. The results can be easily extended to deal with linear uncertain systems, where uncertainties are supposed to belong to convex bounded domains providing an H/sub 2/-guaranteed cost output feedback control. Thanks to the properties of the above-mentioned function, we show that under certain conditions, convex programming tools can be used for numerical purposes. Examples illustrate the theoretical results.

Modified Lagrangians and Monotone Maps in Optimization

Abstract: Introduction to Convex Analysis Modified Lagrangian Functions for Convex Programming Problems Dual Methods Monotone Maps Gradient-Type Methods and Modification of a Monotone Map Saddle Gradient Methods Modified Lagrangian Functions for Smooth Mathematical Programming Problems and Related Dual Methods Bibliographic Comments References Index

Design of fuzzy control systems based on relaxed LMI stability conditions

Abstract: This paper presents new stability conditions and LMI (linear matrix inequality) based designs for both continuous and discrete fuzzy control systems. First, Takagi and Sugeno's fuzzy models and some stability results are recalled. To design fuzzy control systems, nonlinear systems are represented by Takagi-Sugeno fuzzy models. The concept of parallel distributed compensation is employed to design fuzzy controllers from the Takagi-Sugeno fuzzy models. New stability conditions are obtained by relaxing the stability conditions derived in previous papers. The stability analysis of the feedback system is reduced to a problem of finding a common Lyapunov function for a set of linear matrix inequalities. Convex optimization techniques involving LMIs are utilized to find a common Lyapunov function. A procedure to design the fuzzy controller is constructed using the parallel distributed compensation and the relaxed stability conditions. Some stability issues are remarked. A simple example demonstrates the effects of the derived stability conditions.

A Globally and Superlinearly Convergent Algorithm for Nonsmooth Convex Minimization

Abstract: It is well known that a possibly nondifferentiable convex minimization problem can be transformed into a differentiable convex minimization problem by way of the Moreau--Yosida regularization. This paper presents a globally convergent algorithm that is designed to solve the latter problem. Under additional semismoothness and regularity assumptions, the proposed algorithm is shown to have a Q-superlinear rate of convergence.

A quasi-second-order proximal bundle algorithm.

Abstract: This paper introduces an algorithm for convex minimization which includes quasi-Newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm. The method uses the Hessian of a certain outer function which depends on the Jacobian of a proximal point mapping which, in turn, depends on the preconditioner matrix and on a Lagrangian Hessian relative to a certain tangent space. Convergence is proved under boundedness assumptions on the preconditioner sequence.

Barycentric scenario trees in convex multistage stochastic programming

Abstract: This work deals with the approximation of convex stochastic multistage programs allowing prices and demand to be stochastic with compact support. Based on earlier results, sequences of barycentric scenario trees with associated probability trees are derived for minorizing and majorizing the given problem. Error bounds for the optimal policies of the approximate problem and duality analysis with respect to the stochastic data determine the scenarios which improve the approximation. Convergence of the approximate solutions is proven under the stated assumptions. Preliminary computational results are outlined.

Optimal Redesign of Linear Systems

Abstract: This paper proposes a redesign procedure for linear systems. We suppose that an initial satisfactory controller which yields the desired performance is given. Then both the plant and the controller are redesigned to minimize the required active control effort. Either the closed-loop system matrix or the closed-loop covariance matrix of the initial design can be preserved under the redesign. Convex quadratic programming solves this problem. In addition, an iterative approach for integrated plant and controller design is proposed, which uses the above optimal plant/controller redesign in each iterative step. The algorithm has guaranteed convergence and provides a sequence of designs with monotonically decreasing active control effort. Examples are included to illustrate the procedure.

On constrained infinite-time linear quadratic optimal control

Abstract: This work presents a solution to the infinite-time linear quadratic optimal control (ITLQOC) problem with state and control constraints. It is shown that a single, finite dimensional, convex program of known size can yield this solution. Properties of the resulting value function, with respect to initial conditions, are also established and are shown to be useful in determining the aforementioned problem size. An example illustrating the method is finally presented.

On augmented Lagrangian decomposition methods for multistage stochastic programs

Abstract: A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios and by decomposing it into nodes corresponding to stages. Theoretical convergence properties of the two approaches are derived and a computational illustration is presented.

An LMI-based stable fuzzy control of nonlinear systems and its application to control of chaos

Abstract: We present a systematic framework for the stability and design of nonlinear fuzzy control systems. First we represent a nonlinear plant with a Takagi-Sugeno fuzzy model. Then a model-based fuzzy controller design utilizing the concept of so-called "parallel distributed compensation" is employed. The main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system. The design procedure is conceptually simple and natural. Moreover, the stability analysis and control design problems can be reduced to linear matrix inequality (LMI) problems. Therefore they can be solved efficiently in practice by convex programming techniques for LMIs. The design methodology is illustrated by application to the problem of modeling and control of a chaotic system-Chua's circuit.

The efficiency of subgradient projection methods for convex optimization, part I: general level methods

Abstract: We study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value. We present several variants and show that they enjoy almost optimal efficiency estimates. In another paper we discuss possible implementations of such methods. In particular, their projection subproblems may be solved inexactly via relaxation methods, thus opening the way for parallel implementations. They can also exploit accelerations of relaxation methods based on simultaneous projections, surrogate constraints, and conjugate and projected (conditional) subgradient techniques.

Robustness Analysis of Uncertain and Nonlinear Systems

Abstract: Control design is often done based on simplified models. After design it is necessary to verify that the real closed loop system behaves well. This is mostly done by experiments and simulations. Theoretical analysis is an important complement to this that can help to verify critical cases. Structural information about uncertainties, time-variations, nonlinearities, and signals can be described by integral quadratic constraints. The information provided by these constraints can be used to reduce conservatism in analysis of robust stability and robust performance. This thesis treats several aspects of this method for robustness analysis. It is shown how the Popov criterion can be used in combination with other integral quadratic constraints. A new Popov criterion for systems with slowly time-varying polytopic uncertainty is obtained as a result of this. A corresponding result for systems with parametric uncertainty is also derived. The robustness analysis is in practice a problem of finding the most appropriate integral quadratic constraint. This can be formulated as a convex but infinite-dimensional optimization problem. The thesis introduces a flexible format for computations over finite-dimensional subspaces. The restricted optimization problem can generally be formulated in terms of linear matrix inequalities. Duality theory is used to obtain bounds on the computational conservatism. A class of problems is identified for which the dual is particularly attractive. (Less)