André L. Tits

Bio: André L. Tits is an academic researcher from University of Maryland, College Park. The author has contributed to research in topics: Optimization problem & Singular value. The author has an hindex of 32, co-authored 115 publications receiving 4759 citations. Previous affiliations of André L. Tits include University of California, Berkeley & Lund University.

Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics

Abstract: Continuing the development of the structured singular value approach to robust control design, the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty. The problem is shown to be equivalent to a smooth constrained finite-dimensional optimization problem. In view of the fact that the functional to be maximized may have several local extrema, an upper bound on mu whose computation is numerically tractable is established; this leads to a sufficient condition of robust stability and performance. A historical perspective on the development of the mu theory is included. >

User's Guide for CFSQP Version 2.0: A C Code for Solving (Large Scale) Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints

Abstract: CFSQP is a set of C functions for the minimization of the maximum of a set of smooth objective functions (possibly a single one, or even none at all) subject to general smooth constraints (if there is no objective function, the goal is to simply nd a point satisfying the constraints). If the initial guess provided by the user is infeasible for some inequality constraint or some linear equality constraint, CFSQP rst generates a feasible point for these constraints; subsequently the successive iterates generated by CFSQP all satisfy these constraints. Nonlinear equality constraints are turned into inequality constraints (to be satis ed by all iterates) and the maximum of the objective functions is replaced by an exact penalty function which penalizes nonlinear equality constraint violations only. When solving problems with many sequentially related constraints (or objectives), such as discretized semiin nite programming (SIP) problems, CFSQP gives the user the option to use an algorithm that e ciently solves these problems, greatly reducing computational e ort. The user has the option of either requiring that the objective function (penalty function if nonlinear equality constraints are present) decrease at each iteration after feasibility for nonlinear inequality and linear constraints has been reached (monotone line search), or requiring a decrease within at most four iterations (nonmonotone line search). He/She must provide functions that de ne the objective functions and constraint functions and may either provide functions to compute the respective gradients or require that CFSQP estimate them by forward nite di erences. CFSQP is an implementation of two algorithms based on Sequential Quadratic Programming (SQP), modi ed so as to generate feasible iterates. In the rst one (monotone line search), a certain Armijo type arc search is used with the property that the step of one is eventually accepted, a requirement for superlinear convergence. In the second one the same e ect is achieved by means of a onmonotone" search along a straight line. The merit function used in both searches is the maximum of the objective functions if there is no nonlinear equality constraints, or an exact penalty function if nonlinear equality constraints are present. Conditions for External Use 1. The CFSQP routines may not be distributed to third parties. Interested parties should contact the authors directly. 2. If modi cations are performed on the routines, these modi cations will be communicated to the authors. The modi ed routines will remain the sole property of the authors. 3. Due acknowledgment must be made of the use of the CFSQP routines in research reports or publications. Whenever such reports are released for public access, a copy should be forwarded to the authors. 4. The CFSQP routines may only by used for research and development, unless it has been agreed otherwise with the authors in writing. User's Guide for CFSQP Version 2.5 (Released April 1997) Copyright c 1993-1997 by Craig T. Lawrence, Jian L. Zhou, and Andr e L. Tits All Rights Reserved. Enquiries should be directed to Prof. Andr e L. Tits Electrical Engineering Dept. and Institute for Systems Research University of Maryland College Park, Md 20742 U. S. A. Phone : 301-405-3669 Fax : 301-405-6707 E-mail : andre@eng.umd.edu 1 Contents 1 Introduction 2 2 Description of the Basic Algorithms 4 3 Re nements for the Case of Many Objectives/Constraints 12 4 Speci cation of CFSQP 16 5 User-Accessible Stopping Criterion and Flags 22 6 Description of the Output 23 7 User-Supplied Functions 27 7.1 Function obj() : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 7.2 Function constr() : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 7.3 Function gradob() : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 7.4 Function gradcn() : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 8 Organization of CFSQP and Main Functions 31 8.1 Main Functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 8.2 Other Functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 32 9 Examples 33 10 Results for Test Problems 59 11 Programming Tips 61 12 Portability 62 13 Trouble-Shooting 62 14 Acknowledgments 63 15 References 63 2 1 Introduction CFSQP (C code for Feasible Sequential Quadratic Programming) is a set of C functions for the minimization of the maximum of a set of smooth objective functions (possibly a single one, or even none at all) subject to nonlinear equality and inequality constraints, linear equality and inequality constraints, and simple bounds on the variables. In addition, CFSQP contains special provisions for e ciently handling problems with many sequentially related objectives/constraints, for example discretized Semi-In nite Programming (SIP) problems. In the case when no sequentially related constraints or objectives are present, CFSQP tackles optimization problems of the form (P ) minimize max i2If ffi(x)g s.t. x 2 X where If = f1; : : : ; nfg (If = ; if nf = 0) and X is the set of points x 2 IRn satisfying bl x bu gj(x) 0; j = 1; : : : ; ni gj(x) hcj ni; xi dj ni 0; j = ni + 1; : : : ; ti hj(x) = 0; j = 1; : : : ; ne hj(x) haj ne; xi bj ne = 0; j = ne + 1; : : : ; te with bl 2 IRn; bu 2 IRn; fi : IRn ! IR; i = 1; : : : ; nf smooth; gj : IRn ! IR; j = 1; : : : ; ni nonlinear and smooth; cj 2 IRn, dj 2 IR, j = 1; : : : ; ti ni; hj : IRn ! IR; j = 1; : : : ; ne nonlinear and smooth; aj 2 IRn, bj 2 IR, j = 1; : : : ; te ne. Note that it is allowed to have nf = 0, in which case problem (P ) is one of nding a point that satis es a given set of constraints. In full generality, i.e. including sequentially related objectives and constraints, CFSQP handles problems of the form (Psr) minimize maxfmax i2If fi(x);max i2Isr max !2 fi fi(x; !)g s.t. x 2 X where If = f1; : : : ; nf nfsrg (If = ; if nf = 0), Isr = fnf nfsr + 1; : : : ; nfg (Isr = ; if nfsr = 0), fi is an index set for objective functions that are somehow sequentially related, and X is now the set of points x 2 IRn satisfying bl x bu gj(x) 0; j = 1; : : : ; ni nsr gj(x; ) 0; 8 2 gj ; j = ni nsr + 1; : : : ; ni gj(x) hcj ni; xi dj ni 0; j = ni + 1; : : : ; ti `sr gj(x; ) hcj ni( ); xi dj ni( ) 0; 8 2 gj ; j = ti `sr + 1; : : : ; ti hj(x) = 0; j = 1; : : : ; ne hj(x) haj ne; xi bj ne = 0; j = ne + 1; : : : ; te 3 with bl 2 IRn; bu 2 IRn; fi : IRn ! IR; i 2 If smooth; fi : IRn fi ! IR; i 2 Isr continuously di erentiable with respect to the rst argument for each ! 2 fi ; gj : IRn ! IR; j = 1; : : : ; ni nsr nonlinear and smooth; gj : IRn gj ! IR; j = ni nsr +1; : : : ; ni nonlinear, continuously di erentiable with respect to the rst argument for each 2 gj ; cj ni 2 IRn, dj ni 2 IR, j = ni+1; : : : ; ti `sr; cj ni : gj ! IRn, dj ni : gj ! IR; j = ti `sr+1; : : : ; ti; hj : IRn ! IR; j = 1; : : : ; ne nonlinear and smooth; aj 2 IRn, bj 2 IR, j = 1; : : : ; te ne. From this point forward, in order to ease the presentation of the algorithm, we discuss problem (P ), postponing discussion of the algorithm used to solve (Psr) until x 3. If the initial guess provided by the user is infeasible for linear constraints, CFSQP generates a point satisfying these constraints by solving a strictly convex quadratic program (QP). Next, if the initial guess, or the newly generated initial guess, is infeasible for the nonlinear inequality constraints, CFSQP generates a point x0 satisfying all constraints (other than nonlinear equality constraints) by iterating on the problem of minimizing the maximum of the nonlinear inequality constraints. Then, using a scheme due to Mayne and Polak [1] and adapted to the FSQP framework in [2], nonlinear equality constraints are turned into inequality constraints2 hj(x) 0; j = 1; : : : ; ne and the original objective function maxi2Ifffi(x)g is replaced by the modi ed objective function fm(x; p) = max i2If ffi(x)g ne X j=1 pjhj(x); where pj, j = 1; : : : ; ne, are positive penalty parameters that are iteratively adjusted. If nf = 0, the \max" is de ned to be identically zero. The resulting optimization problem therefore involves only linear constraints and nonlinear inequality constraints. The successive iterates generated by CFSQP all satisfy these constraints. The user has the option of either requiring that the exact penalty function (the maximum value of the objective functions if no nonlinear equality constraints are present) decrease at each iteration (after feasibility for original nonlinear inequality and linear constraints has been reached), or requiring a decrease within at most four iterations. He/She must provide functions that de ne the objectives and constraints, and may either provide functions to compute the respective gradients or require that CFSQP estimate them by forward nite di erences. Thus, CFSQP solves the original problem with nonlinear equality constraints by solving a modi ed optimization problem with only linear constraints and nonlinear inequality constraints. For the transformed problem, it implements algorithms that are described and analyzed in [3], [4], [5], [6] and [7], with some additional re nements. These algorithms are based on a Sequential Quadratic Programming (SQP) iteration, modi ed so as to generate feasible iterates. The merit function is the objective function. An Armijo-type line search 2For every j for which hj(x0) > 0, \hj(x) = 0" is rst replaced by \ hj(x) = 0" and hj is renamed hj . 4 is used (along the arc described below for the monotone line search) when minimizing the maximum of the nonlinear inequality constraints to generate an initial feasible point. After obtaining feasibility, either (i) an Armijo-type line search may be used, yielding a monotone decrease of the objective function at each iteration [3]; or (ii) a nonmonotone line search (inspired from [8] and analyzed in [4] and [5] in the present context) may be selected, forcing a decreas

DELIGHT.SPICE: an optimization-based system for the design of integrated circuits

Abstract: DELIGHT.SPICE is the union of the DELIGHT interactive optimization-based computer-aided-design system and the SPICE circuit analysis program. With the DELIGHT.SPICE tool, circuit designers can take advantage of recent powerful optimization algorithms and a methodology that emphasizes designer intuition and man-machine interaction. Designer and computer are complementary in adjusting parameters of electronic circuits automatically to improve their performance criteria and to study complex tradeoffs between multiple competing objectives, while simultaneously satisfying multiple-constraint specifications. The optimization runs much more efficiently than previously because the SPICE program used has been enhanced to perform DC, AC, and transient sensitivity computation. Industrial analog and digital circuits have been redesigned using this tool, yielding substantial improvement in circuit performance. >

A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm

Abstract: A sequential quadratic programming (SQP) algorithm generating feasible iterates is described and analyzed. What distinguishes this algorithm from previous feasible SQP algorithms proposed by various authors is a reduction in the amount of computation required to generate a new iterate while the proposed scheme still enjoys the same global and fast local convergence properties. A preliminary implementation has been tested and some promising numerical results are reported.

On combining feasibility, descent and superlinear convergence in inequality constrained optimization

Abstract: Extension of quasi-Newton techniques from unconstrained to constrained optimization via Sequential Quadratic Programming (SQP) presents several difficulties. Among these are the possible inconsistency, away from the solution, of first order approximations to the constraints, resulting in infeasibility of the quadratic programs; and the task of selecting a suitable merit function, to induce global convergence. In ths case of inequality constrained optimization, both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints, and that yield monotonically decreasing objective function values. (Feasibility of the successive iterates is in fact required in many contexts such as in real-time applications or when the objective function is not well defined outside the feasible set.) It has been recently shown that this can be achieved while preserving local two-step superlinear convergence. In this note, the essential ingredients for an SQP-based method exhibiting the desired properties are highlighted. Correspondingly, a class of such algorithms is described and analyzed. Tests performed with an efficient implementation are discussed.

Numerical Optimization

Abstract: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Linear Matrix Inequalities in System and Control Theory

Abstract: Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming

Abstract: We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.

Stability of Time-Delay Systems

Abstract: Preface, Notations 1.Introduction to Time-Delay Systems I.Frequency-Domain Approach 2.Systems with Commensurate Delays 3.Systems withIncommensurate Delays 4.Robust Stability Analysis II.Time Domain Approach 5.Systems with Single Delay 6.Robust Stability Analysis 7.Systems with Multiple and Distributed Delays III.Input-Output Approach 8.Input-output stability A.Matrix Facts B.LMI and Quadratic Integral Inequalities Bibliography Index