On limited-memory quasi-Newton methods for minimizing a quadratic function

The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give two classes of limited-memory quasi-Newton Hessian approximations that generate search directions parallel to those of the method of preconditioned conjugate gradients, and hence give finite termination on quadratic optimization problems. The Hessian approximations are described by a novel compact representation which provides a dynamical framework. We also discuss possible extensions of these classes and show their behavior on randomly generated quadratic optimization problems. The methods behave numerically similar to L-BFGS. Inclusion of information from the first iteration in the limited-memory Hessian approximation and L-BFGS significantly reduces the effects of round-off errors on the considered problems. In addition, we give our compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components

Solving the Optimal Mistuning Problem by Symmetry: A General Framework for Extending Flutter Boundaries in Turbomachines via Mistuning

A general framework is presented for analyzing and optimizing stability increases due to mistuning. The framework given is model independent and is based primarily on symmetry arguments. Difficult practical issues are transformed to tractable mathematical questions. It is shown that mistuning analysis reduces to a block circular matrix eigenvalue/vector problem which can be solved efficiently even for large problems. Similarly, the optimization becomes a standard linear constraint quadratic programming problem and can be solved numerically. Since the methods given are model independent, they can be applied to various models and allow the researcher to easily conclude which models accurately capture mistuning, and which do not. A simple quasi-steady model for flutter in a cascade is used to illustrate and validate results in this paper

An exact solution method for binary equilibrium problems with compensation and the power market uplift problem

We propose a novel method to find Nash equilibria in games with binary decision variables by including compensation payments and incentive-compatibility constraints from non-cooperative game theory directly into an optimization framework in lieu of using first order conditions of a linearization, or relaxation of integrality conditions. The reformulation offers a new approach to obtain and interpret dual variables to binary constraints using the benefit or loss from deviation rather than marginal relaxations. The method endogenizes the trade-off between overall (societal) efficiency and compensation payments necessary to align incentives of individual players. We provide existence results and conditions under which this problem can be solved as a mixed-binary linear program. We apply the solution approach to a stylized nodal power-market equilibrium problem with binary on-off decisions. This illustrative example shows that our approach yields an exact solution to the binary Nash game with compensation. We compare different implementations of actual market rules within our model, in particular constraints ensuring non-negative profits (no-loss rule) and restrictions on the compensation payments to non-dispatched generators. We discuss the resulting equilibria in terms of overall welfare, efficiency, and allocational equity

Limited-memory BFGS Systems with Diagonal Updates

In this paper, we investigate a formula to solve systems of the form (B + {\sigma}I)x = y, where B is a limited-memory BFGS quasi-Newton matrix and {\sigma} is a positive constant. These types of systems arise naturally in large-scale optimization such as trust-region methods as well as doubly-augmented Lagrangian methods. We show that provided a simple condition holds on B_0 and \sigma, the system (B + \sigma I)x = y can be solved via a recursion formula that requies only vector inner products. This formula has complexity M^2n, where M is the number of L-BFGS updates and n >> M is the dimension of x