At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which ∂f(p) has 0-breadth). This property opens the way to defining a suitably restricted second derivative of f at p. We do this via an intermediate function, convex on U . We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.

/pdf/the-lagrangian-of-a-convex-function-1ih0y8y5uk.pdf

The -Lagrangian of a convex function

We take up the question of second-order expansions for a class of functions of importance in optimization, namely, Moreau envelope regularizations of nonsmooth functions $f$. It is shown that when $f$ is prox-regular, which includes convex functions and the extended real-valued functions representing problems of nonlinear programming, the many second-order properties that can be formulated around the existence and stability of expansions of the envelopes of $f$ or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of $f$ itself. This clarifies the circumstances conducive to developing computational methods based on envelope functions, such as second-order approximations in nonsmooth optimization and variants of the proximal point algorithm. The results establish that generalized second-order expansions of Moreau envelopes, at least, can be counted on in most situations of interest in finite-dimensional optimization.

Generalized Hessian Properties of Regularized Nonsmooth Functions

The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on Rin and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in another. For this case a concept of epi-hypo-differentiability is introduced. The saddle function results provide a foundation for the sensitivity analysis of primal and dual optimal solutions to general finite-dimensional problems in convex optimization, since such solutions are characterized as saddlepoints of a convex-concave Lagrangian function, or equivalently as subgradients of the saddle function conjugate to the Lagrangian.

/pdf/generalized-second-derivatives-of-convex-functions-and-3i18vhmtfn.pdf

Generalized second derivatives of convex functions and saddle functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C1,1 functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gateaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.

Generalized second-order directional derivatives and optimization with C1,1 functions

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Second-order global optimality conditions for convex composite optimization

In this work a new notion of generalized second-order directional derivative and generalized Hessian for nonsmooth real-valued functions is studied. The general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation.Two applications of these derivatives in optimization theory are considered: first, to obtaining necessary and sufficient second-order optimality conditions for problems with or without constraints; and second, to extending the Newton method for the minimization of a $\mathcal{C}^{1,1} $ function.

A generalized second-order derivative in nonsmooth optimization

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

R. Correa

Author Tools

Chat about Author

Papers

A generalized second-order derivative in nonsmooth optimization