A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles

Abstract

We propose an algorithm for solving nonsmooth, nonconvex, constrained optimization problems as well as a new set of visualization tools for comparing the performance of optimization algorithms. Our algorithm is a sequential quadratic optimization method that employs Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton Hessian approximations and an exact penalty function whose parameter is controlled using a steering strategy. While our method has no convergence guarantees, we have found it to perform very well in practice on challenging test problems in controller design involving both locally Lipschitz and non-locally-Lipschitz objective and constraint functions with constraints that are typically active at local minimizers. In order to empirically validate and compare our method with available alternatives-on a new test set of 200 problems of varying sizes-we employ new visualization tools which we call relative minimization profiles. Such profiles are designed to simultaneously assess the relative performance of several algorithms with respect to objective quality, feasibility, and speed of progress, highlighting the trade-offs between these measures when comparing algorithm performance.

A rational H∞‐norm–based approach for the optimal design of seismically excited reinforced concrete frames

“…First of all, the spectra should be expressed in terms of the frequency variable . To derive a transfer function representation G f (i ) of the piecewise defined spectra in Equation 23, zeros and poles should first be listed and classified, ie, so as to generate a critical frequency vector that groups zeros and poles that, for the case at hand, reads Secondly, a transfer function representation G f (s) of the frequency-dependent spectra should be obtained to be used in Equation 7.…”

“…For the sake of this paper, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) Sequential Quadratic Programming (SQP) method for nonsmooth, nonconvex, constrained optimization developed in Curtis et al 23 has been used as a viable alternative to the Method of Moving Asymptotes (MMA) 24 that works fine as well for minimum H ∞ -norm problems; see, eg, Venini and Pingaro. 14,15 To address the nonconvexity issue, all numerical optimization problems to be presented next were restarted from different initial designs, and convergence to the very same solution was observed.…”

“…Secondly, a transfer function representation G f (s) of the frequency-dependent spectra should be obtained to be used in Equation 7. To start with, some algebra allows to rewrite EC8 acceleration spectra with respect to the Laplace variable s as follows: Figure 4 shows the graphs of the frequency-dependent spectra given in Equation 23. To derive a transfer function representation G f (i ) of the piecewise defined spectra in Equation 23, zeros and poles should first be listed and classified, ie, so as to generate a critical frequency vector that groups zeros and poles that, for the case at hand, reads…”

“…Such last zero may be written as B where > 1 is a real number to be chosen, eg, by minimizing the L 2 error between the actual spectrum in Equation23 and its approximation G f (s). The transfer function is no longer constant, and in fact, B is a pole for G f (s) that ensures an overall decrease rate of −20 dB/decade.…”

Trade‐off between energy consumption and ethylene recovery rate for quasi‐plant wide operation optimization of the ethylene column with bottom circulatory system in ethylene complex

“…Considering the large number of equations, the advanced solving method, such as Broyden-Fletcher-Goldfarb-Shanno SQP method [40] and interior point method, [41] should be employed. But the optimizer of gPROMS, named as SRQPD solver, employs SQP method for the solution of the nonlinear programming problem.…”