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Convex and quasiconvex truncations of nonconvex functions

Cornel Pintea

Abstract

We consider nonconvex real valued functions whose truncations are either quasiconvex or even convex starting with a certain level. Among them, the $C^2$-smooth functions whose level sets are all completely contained in the positive definite region of their Hessian matrices, starting with a certain level, are good examples of such functions. For such a function we show the injectivity of its restricted gradient to a large subset of the positive definite region of its Hessian matrices.

Convex and quasiconvex truncations of nonconvex functions

Abstract

We consider nonconvex real valued functions whose truncations are either quasiconvex or even convex starting with a certain level. Among them, the -smooth functions whose level sets are all completely contained in the positive definite region of their Hessian matrices, starting with a certain level, are good examples of such functions. For such a function we show the injectivity of its restricted gradient to a large subset of the positive definite region of its Hessian matrices.
Paper Structure (5 sections, 9 theorems, 72 equations, 1 figure)

This paper contains 5 sections, 9 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Convex and quasiconvex truncations
  3. The ${\rm Hess}^+$ region towards truncation convexity
  4. The injectivity of the gradient on a large overlevel set
  5. Final remarks and open questions

Key Result

Proposition 2.1

If $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ is truncated convex and lower semicontinuous, then $T_{\rm scl}(f)$ is convex.

Figures (1)

  • Figure 1:

Theorems & Definitions (29)

  • Definition 2.1
  • Example 2.1
  • Remark 2.1
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • proof : Proof of Proposition \ref{['theSCLachived']}
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • ...and 19 more