Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

  title={Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization},
  author={Pablo A. Parrilo},
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… 

Exact relaxation for the semidefinite matrix rank minimization problem with extended Lyapunov equation constraint

A special semidefinite matrix rank minimization problem with the extended Lyapunov equation constraint arising from low-order optimal control is considered and shown to possess the desired exact relaxation properties by exploiting the special structures of the involved linear transformation.

Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the

Semidefinite programming in matrix unknowns which are dimension free

One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the

Generic Properties for Semialgebraic Programs

In this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints, in which every problem of the class is obtained by linear

Stability and Genericity for Semi-algebraic Compact Programs

It is shown that, for almost every problem in the class of polynomial optimization problems over semi-algebraic compact sets, there is a unique optimal solution for which the global quadratic growth condition and the strong second-order sufficient conditions hold.

LMI Approximations for Cones of Positive Semidefinite Forms

This work derives a novel approximation to the cone of copositive forms, that is, the cones of forms that are positive semidefinite over the nonnegative orthant, which opens the possibility of using linear programming technology in optimization problems over these cones.

Global optimization of rational functions: a semidefinite programming approach

It is shown that in the univariate case (n = 1), these problems have exact reformulations as semidefinite programming (SDP) problems, by using reformulations introduced in the PhD thesis of Jibetean [16].

Stable Rank-One Matrix Completion is Solved by the Level 2 Lasserre Relaxation

It is shown that in every instance where the problem has a unique solution, one can provably recover the original matrix through the level 2 Lasserre relaxation with minimization of the trace norm.

Convex algebraic geometry and semidefinite optimization

  • P. Parrilo
  • Mathematics, Computer Science
    ISSAC '13
  • 2013
This tutorial will focus on basic and recent developments in convex algebraic geometry, and the associated computational methods based on semidefinite programming for optimization problems involving polynomial equations and inequalities.

Computing optimal fixed order H∞-synthesis values by matrix sum of squares relaxations

  • C. HolC. Scherer
  • Mathematics, Computer Science
    2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601)
  • 2004
Two approaches to solve robust LMI problems based on sos matrix decompositions, a direct approach and one based on the S-procedure are presented, which lead to an asymptotically exact family of LMI relaxations for computing lower bounds on the optimal fixed-order H∞-norm whose size only grow quadratically in the dimension of the system state.



A bisection method for computing the H∞ norm of a transfer matrix and related problems

The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise.

Semidefinite Programming

A survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution are given.

On the rank minimization problem over a positive semidefinite linear matrix inequality

This paper considers the problem of minimizing the rank of a positive semidefinite matrix, subject to the constraint that an affine transformation of it is also positive semidfinite, and employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice.

A linear matrix inequality approach to H∞ control

The continuous- and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach:

Nonlinear optimal control: an enhanced quasi-LPV approach

Realistic models of physical systems are often nonlinear. Our objective is to synthesize controllers for nonlinear systems that not only provide stability, but also deliver good closed-loop

A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems

It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method, and a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions.

A hierarchy of relaxation between the continuous and convex hull representations

In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into

Least squares stationary optimal control and the algebraic Riccati equation

The optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated. Both the case in which the terminal state is free and that in which the

Output feedback stabilization and related problems-solution via decision methods

This paper shows how this and a number of other linear system theory problems can be simply reformulated so as to allow application of known algorithms for solution of the existence question, with the construction problem being solved by some extension of these known algorithms.

On the stability domain estimation via a quadratic Lyapunov function: convexity and optimality properties for polynomial systems

This estimate of the stability domain of the origin of an n-order polynomial system can be obtained by solving a suitable convex optimization problem and is shown to be optimal for an important subclass including both quadratic and cubic systems.