On generalized gradients and optimization

نویسنده

  • Erik J. Balder
چکیده

There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus for convex nondifferentiable functions [1]. This development started with F.H. Clarke, who introduced a generalized gradient for functions that are locally Lipschitz, but (possibly) nondifferentiable. Generalized gradients turn out to be the subdifferentials, in the sense of convex analysis [1], of generalized directional derivative functions that are canonically associated to the locally Lipschitz functions under consideration. The key point to note is that such generalized directional derivative functions are automatically convex, even when the original locally Lipschitz functions are not convex. Two important special cases of functions f that are locally Lipschitz near some point x0 ∈ R and their generalized gradients, denoted by ∂̄f(x0), are as follows: (i) f is continuously differentiable at x0 and (ii) f is convex on R and x0 ∈ int dom f . Then

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Optimization of thermal curing cycle for a large epoxy model

Heat generation in an exothermic reaction during the curing process and low thermal conductivity of the epoxy resin produces high peak temperature and temperature gradients which result in internal and residual stresses, especially in large epoxy samples. In this paper, an optimization algorithm was developed and applied to predict the thermal cure cycle to minimize the temperature peak and the...

متن کامل

Lagrange Multipliers for Nonconvex Generalized Gradients with Equality, Inequality and Set Constraints

A Lagrange multiplier rule for nite dimensional Lipschitz problems is proven that uses a nonconvex generalized gradient. This result uses either both the linear generalized gradient and the generalized gradient of Mordukhovich or the linear generalized gradient and a qualiication condition involving the pseudo-Lipschitz behavior of the feasible set under perturbations. The optimization problem ...

متن کامل

The Linear Nonconvex Generalized Gradient and Lagrange Multipliers

A Lagrange multiplierrules that uses small generalized gradients is introduced. It includes both inequality and set constraints. The generalized gradient is the linear generalized gradient. It is smaller than the generalized gradients of Clarke and Mordukhovich but retains much of their nice calculus. Its convex hull is the generalized gradient of Michel and Penot if a function is Lipschitz. Th...

متن کامل

GENERALIZED FLEXIBILITY-BASED MODEL UPDATING APPROACH VIA DEMOCRATIC PARTICLE SWARM OPTIMIZATION ALGORITHM FOR STRUCTURAL DAMAGE PROGNOSIS

This paper presents a new model updating approach for structural damage localization and quantification. Based on the Modal Assurance Criterion (MAC), a new damage-sensitive cost function is introduced by employing the main diagonal and anti-diagonal members of the calculated Generalized Flexibility Matrix (GFM) for the monitored structure and its analytical model. Then, ...

متن کامل

SIZE AND GEOMETRY OPTIMIZATION OF TRUSS STRUCTURES USING THE COMBINATION OF DNA COMPUTING ALGORITHM AND GENERALIZED CONVEX APPROXIMATION METHOD

In recent years, the optimization of truss structures has been considered due to their several applications and their simple structure and rapid analysis. DNA computing algorithm is a non-gradient-based method derived from numerical modeling of DNA-based computing performance by new computers with DNA memory known as molecular computers. DNA computing algorithm works based on collective intelli...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008