Convex Composite Function Research Articles

Sharp Variational Conditions for Convex Composite Nonsmooth Functions

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistoryPublished online: 28 July 2006KeywordsGâteaux approximate Jacobians, sharp necessary conditions, convex composite functions, second-order variational conditionsAMS Subject Headings49A52, 90C30, 26A24Publication DataISSN (print): 1052-6234ISSN (online): 1095-7189Publisher: Society for Industrial and Applied MathematicsCODEN: sjope8

More on the horizon function of a convex composite function

We introduce a suitable definition for the horizon function of a convex mapping with values in a Daniell space and prove that the horizon function of a convex composite function coincides with the composite of the corresponding horizon functions.

A general formula for the horizon function of a convex composite function

In many variational problems the behavior of functions or sets at infinity is of crucial importance. An appropriate concept for dealing with this topic has been introduced and widely studied under the name of recession analysis. The terminology is not unique: the recession cones (resp. functions) are also called asymptotic cones (resp. functions) or horizon cones (resp. functions). In the convex case calculus rules are available for the horizon function of a sum or a max of two functions, of a marginal function, of the composite of a convex function with an affine operator. The purpose of this paper is to provide a general formula for the composite of an extended real-valued convex function with a convex mapping with values in an ordered topological vector space. It seems that this problem has never been tackled in full generality. Some preliminaries about lower semicontinuous mappings and convex composite functions are necessary. Various examples are given.

Convex composite functions in Banach spaces and the primal lower-nice property

Primal lower-nice functions defined on Hilbert spaces provide examples of functions that are “integrable” (i.e. of functions that are determined up to an additive constant by their subgradients). The class of primal lower-nice functions contains all convex and lower- C 2 C^2 functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a C 2 C^2 mapping under a constraint qualification. In Banach spaces certain convex composite functions were known to be primal lower-nice (e.g. a convex function had to be continuous relative to its domain). In this paper we weaken the assumptions and provide new examples of convex composite functions defined on a Banach space with the primal lower-nice property. One consequence of our results is the identification of new examples of integrable functions on Hilbert spaces.

Evolution equations governed by the subdifferential of a convex composite function in finite dimensional spaces

Under quite general assumptions, we prove existence, uniqueness and regularity of a solution $U$ to the evolution equation $-U'(t)\in\partial(g\circ F)(U(t))$, $U(0)=u_0$, where $g:\mathbb{R}^q\rightarrow\mathbb{R}\cup\{+\infty\}$ is a closed proper convex function, $F:\mathbb{R}^p\rightarrow \mathbb{R}^q$ is a continuously differentiable mapping whose gradient is Lipschitz continuous on bounded subsets and $u_0\in\dom (g\circ F)$. We also study the asymptotic behavior of $U$ and give an application to nonlinear mathematical programming.

Solving mixed integer nonlinear programs by outer approximation

A wide range of optimization problems arising from engineering applications can be formulated as Mixed Integer NonLinear Programming problems (MINLPs). Duran and Grossmann (1986) suggest an outer approximation scheme for solving a class of MINLPs that are linear in the integer variables by a finite sequence of relaxed MILP master programs and NLP subproblems. Their idea is generalized by treating nonlinearities in the integer variables directly, which allows a much wider class of problem to be tackled, including the case of pure INLPs. A new and more simple proof of finite termination is given and a rigorous treatment of infeasible NLP subproblems is presented which includes all the common methods for resolving infeasibility in Phase I. The worst case performance of the outer approximation algorithm is investigated and an example is given for which it visits all integer assignments. This behaviour leads us to include curvature information into the relaxed MILP master problem, giving rise to a new quadratic outer approximation algorithm. An alternative approach is considered to the difficulties caused by infeasibility in outer approximation, in which exact penalty functions are used to solve the NLP subproblems. It is possible to develop the theory in an elegant way for a large class of nonsmooth MINLPs based on the use of convex composite functions and subdifferentials, although an interpretation for thel 1 norm is also given.

Optimality conditions for non-finite valued convex composite functions

Burke (1987) has recently developed second-order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued. In this note we present a technique for reducing the infinite valued case to the finite valued one. We then use this technique to extend the results in Burke (1987) to the case in which the convex function may take infinite values. We conclude by comparing these results with those established by Rockafellar (1989) for the piecewise linear-quadratic case.