Partially Augmented Lagrangian Method for Matrix Inequality Constraints

Noll, Dominikus; Torki, Mounir; Apkarian, Pierre

doi:10.1137/s1052623402413963

Cited by 55 publications

(42 citation statements)

References 31 publications

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“…However in some important application problems, see e.g. [102,100,106], it is helpful to incorporate non convex and nonlinear functions into the model resulting in the above (NLSDP).…”

Section: Nonlinear Semidefinite Programmingmentioning

confidence: 99%

“…It bases on generalized augmented Lagrangians designed for the semidefinite constraint and solves a sequence of unconstrained minimization problems driven by a penalty parameter. There are other approaches for dealing with general NLSDP, for instance, sequential semidefinite programming [46,48,36,56,51,52,131,139,43], bundle methods [104,105], partially augmented Lagrangian approach [10,45,106], interior point trust region [89,90,91], predictor-corrector interior point [64], augmented Lagrangian [106,132], successive linearization [68] and primal-dual interior point methods [142,143,69] among others. There is not a definitive answer to the question of which is the most convenient approach for solving NLSDP in general, which explains the intense research activity going on in this area.…”

Section: Nonlinear Semidefinite Programmingmentioning

confidence: 99%

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Linear and Nonlinear Semidefinite Programming

Bofill

Gómez

2014

Pesqui. Oper.

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ABSTRACT. This paper provides a short introduction to optimization problems with semidefinite constraints. Basic duality and optimality conditions are presented. For linear semidefinite programming some advances by dealing with degeneracy and the semidefinite facial reduction are discussed. Two relatively recent areas of application are presented. Finally a short overview of relevant literature on algorithmic approaches for efficiently solving linear and nonlinear semidefinite programming is provided.

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“…However in some important application problems, see e.g. [102,100,106], it is helpful to incorporate non convex and nonlinear functions into the model resulting in the above (NLSDP).…”

Section: Nonlinear Semidefinite Programmingmentioning

confidence: 99%

Section: Nonlinear Semidefinite Programmingmentioning

confidence: 99%

Linear and Nonlinear Semidefinite Programming

Bofill

Gómez

2014

Pesqui. Oper.

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“…al presented a method for low order H ∞ controller synthesis which relaxes only one of the constraints and is thus called a partially augmented Lagrangian method. In [6] the method is extended to more general robust control than H ∞ controller problems and [7] generalizes the framework to optimization problems with general matrix inequality constraints.…”

Section: Introductionmentioning

confidence: 99%

A partially augmented Lagrangian method for low order H-infinity controller synthesis using rational constraints

Ankelhed

Helmersson

Hansson

2011

IEEE Conference on Decision and Control and European Control Conference

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When designing robust controllers, H-in nity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the (augmented) system, the problem becomes nonconvex and it is relatively hard to solve. These problems become very complex, even when the order of the system is low. The approach used in this work is based on formulating the constraint on the maximum order of the controller as a polynomial (or rational) equation. This equality constraint is added to the optimization problem of minimizing an upper bound on the H-in nity norm of the closed loop system subject to linear matrix inequality (LMI) constraints. The problem is then solved by reformulating it as a partially augmented Lagrangian problem where the equality constraint is put into the objective function, but where the LMIs are kept as constraints. The proposed method is evaluated together with two well-known methods from the literature. The results indicate that the proposed method has comparable performance in most cases, especially if the synthesized controller has many parameters, which is the case if the system to be controlled has many input and output signals. Abstract-When designing robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the (augmented) system, the problem becomes nonconvex and it is relatively hard to solve. These problems become very complex, even when the order of the system is low.The approach used in this work is based on formulating the constraint on the maximum order of the controller as a polynomial (or rational) equation. This equality constraint is added to the optimization problem of minimizing an upper bound on the H-infinity norm of the closed loop system subject to linear matrix inequality (LMI) constraints. The problem is then solved by reformulating it as a partially augmented Lagrangian problem where the equality constraint is put into the objective function, but where the LMIs are kept as constraints.The proposed method is evaluated together with two wellknown methods from the literature. The results indicate that the proposed method has comparable performance in most cases, especially if the synthesized controller has many parameters, which is the case if the system to be controlled has many input and output signals.

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“…In response, various new design methods have been proposed since the late 1990s, including methods from global optimization [6,7,34,37], matrix inequality constrained nonlinear programming [5,14,24,25,36], augmented Lagrangian methods [13,30], eigenvalue optimization [1][2][3]19], and others. Presently we discuss a new strategy for H ∞ output feedback control synthesis, which has several advantages over existing approaches, because it avoids the use of Lyapunov variables.…”

Section: Introductionmentioning

confidence: 99%

Second-order nonsmooth optimization for H ∞ synthesis

Bompart¹,

Noll²,

Apkarian³

2007

Numer. Math.

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The standard way to compute H ∞ feedback controllers uses algebraic Riccati equations and is therefore of limited applicability. Here we present a new approach to the H ∞ output feedback control design problem, which is based on nonlinear and nonsmooth mathematical programming techniques. Our approach avoids the use of Lyapunov variables, and is therefore flexible in many practical situations.

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Partially Augmented Lagrangian Method for Matrix Inequality Constraints

Cited by 55 publications

References 31 publications

Linear and Nonlinear Semidefinite Programming

Linear and Nonlinear Semidefinite Programming

A partially augmented Lagrangian method for low order H-infinity controller synthesis using rational constraints

Second-order nonsmooth optimization for H ∞ synthesis

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