A Bundle Method for a Class of Bilevel Nonsmooth Convex Minimization Problems
نویسنده
چکیده
We consider the bilevel problem of minimizing a nonsmooth convex function over the set of minimizers of another nonsmooth convex function. Standard convex constrained optimization is a particular case in this framework, corresponding to taking the lower level function as a penalty of the feasible set. We develop an explicit bundle-type algorithm for solving the bilevel problem, where each iteration consists of making one descent step for a weighted sum of the upper and lower level functions, after which the weight can be updated immediately. Convergence is shown under very mild assumptions. We note that in the case of standard constrained optimization, the method does not require iterative solution of any penalization subproblems, not even approximately, and does not assume any regularity of constraints (e.g., the Slater condition). We also present some computational experiments for minimizing a nonsmooth convex function over a set defined by linear complementarity constraints.
منابع مشابه
An efficient one-layer recurrent neural network for solving a class of nonsmooth optimization problems
Constrained optimization problems have a wide range of applications in science, economics, and engineering. In this paper, a neural network model is proposed to solve a class of nonsmooth constrained optimization problems with a nonsmooth convex objective function subject to nonlinear inequality and affine equality constraints. It is a one-layer non-penalty recurrent neural network based on the...
متن کاملA smoothing augmented Lagrangian method for solving simple bilevel programs
In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a...
متن کاملComposite proximal bundle method
We consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping. This is a sufficiently rich class that includes max-functions, largest eigenvalue functions, and norm-1 regularized functions. The bundle method uses an oracle that is able to compute separately the function and subgradient information for...
متن کاملOn solving simple bilevel programs with a nonconvex lower level program
In this paper, we consider a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint and the upper level program has a convex set constraint. By using the value function of the lower level program, we reformulate the bilevel program as a single level optimization problem with a nonsmooth inequality constraint and a convex set constra...
متن کاملEmpirical and Theoretical Comparisons of Several Nonsmooth Minimization Methods and Software
The most of nonsmooth optimization methods may be divided in two main groups: subgradient methods and bundle methods. Usually, when developing new algorithms and testing them, the comparison is made between similar kinds of methods. In this report we test and compare both different bundle methods and different subgradient methods as well as some methods which may be considered as hybrids of the...
متن کاملذخیره 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 ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM Journal on Optimization
دوره 18 شماره
صفحات -
تاریخ انتشار 2007