We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f , not necessarily convex. We introduce an inex- act line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a care- ful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke station- ary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.

/pdf/nonsmooth-optimization-via-quasi-newton-methods-14v2jl5o1y.pdf

Nonsmooth optimization via quasi-Newton methods

We consider the design of optimal state feedback gains subject to structural constraints on the distributed controllers. These constraints are in the form of sparsity requirements for the feedback matrix, implying that each controller has access to information from only a limited number of subsystems. The minimizer of this constrained optimal control problem is sought using the augmented Lagrangian method. Notably, this approach does not require a stabilizing structured gain to initialize the optimization algorithm. Motivated by the structure of the necessary conditions for optimality of the augmented Lagrangian, we develop an alternating descent method to determine the structured optimal gain. We also utilize the sensitivity interpretation of the Lagrange multiplier to identify favorable communication architectures for structured optimal design. Examples are provided to illustrate the effectiveness of the developed method.

/pdf/augmented-lagrangian-approach-to-design-of-structured-vth3jmgwri.pdf

Augmented Lagrangian Approach to Design of Structured Optimal State Feedback Gains

https://homepages.laas.fr/henrion/papers/hifoo-haifa.pdf

Multiobjective robust control with HIFOO 2.0

A novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality (BMI) constraints. The key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings. At each iteration of the algorithm the concave part is linearized, leading to a convex subproblem. Applications to various output feedback controller synthesis problems are presented. In these applications, the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality (LMI) constraints. The performance of the algorithm has been benchmarked on the data from the COMPleib library.

Combining Convex–Concave Decompositions and Linearization Approaches for Solving BMIs, With Application to Static Output Feedback

https://hal.archives-ouvertes.fr/hal-01342560/document

From static output feedback to structured robust static output feedback: A survey

This paper presents an extension of the HIFOO toolbox for Matlab for fixed-structure and fixed-order ℌ ∞ controller design to discrete-time controller design. The approach is applied to a restricted complexity controller synthesis problem for an active suspension system.

/pdf/fixed-structure-discrete-time-controller-synthesis-with-2uh72fu0jz.pdf

Fixed-structure discrete-time ℌ ∞ controller synthesis with HIFOO

Multiobjective control design is known to be a difficult problem both in theory and practice. Our approach is to search for locally optimal solutions of a nonsmooth optimization problem that is built to incorporate minimization objectives and constraints for multiple plants. We report on the success of this approach using our public-domain Matlab toolbox HIFOO 2.0, comparing our results with benchmarks in the literature.

Multiobjective Robust Control with HIFOO 2.0

https://www.cims.nyu.edu/~marcm/papers/lss2007.pdf

Fixed-order and structure H∞ control with model based feedforward for elastic web winding systems

We report on our experience with strong stabilization using HIFOO, a toolbox for H-infinity fixed-order controller design. We applied HIFOO to 21 fixed-order stable H-infinity controller design problems in the literature, comparing the results with those published for other methods. The results show that HIFOO often achieves good H-infinity performance with low-order stable controllers, unlike other methods in the literature.

Marc Millstone

Papers

Multiobjective robust control with HIFOO 2.0

Fixed-structure discrete-time ℌ ∞ controller synthesis with HIFOO

Multiobjective Robust Control with HIFOO 2.0

Fixed-order and structure H∞ control with model based feedforward for elastic web winding systems

H infinity strong stabilization via HIFOO, a package for fixed-order controller design.