Such sequences possess very attractive properties that greatly simplify the analysis of their asymptotic behavior.

## Convex Analysis and Optimal Control - MATH4071

The notion of a convex cone, which lies between that of a linear subspace and that of a convex set, is the main topic of this chapter. It has been very fruitful in many branches of nonlinear analysis. For instance, closed convex cones provide decompositions analogous to the well-known orthogonal decomposition based on closed linear subspaces. They also arise naturally in convex analysis in the local study of a convex set via the tangent cone and the normal cone operators, and they are central in the analysis of various extensions of the notion of an interior that will be required in later chapters.

In this chapter, we develop basic results concerning support points, including the Bishop—Phelps theorem and the representation of a nonempty closed convex set as the intersection of the closed half-spaces containing it. Polar sets are also studied.

Convex functions, which lie at the heart of modern optimization, are introduced in this chapter. We study operations that preserve convexity, and the interplay between various continuity properties.

### Annals of Functional Analysis

The theory of convex functions is most powerful in the presence of lower semicontinuity. A key property of lower semicontinuous convex functions is the existence of a continuous affine minorant, which we establish in this chapter by projecting onto the epigraph of the function. In this chapter we present variants of the notion of convexity for functions.

The most important are the weaker notion of quasiconvexity, and the stronger notions of uniform and strong convexity. Convex optimization is one of the main areas of application of convex analysis. This chapter deals with the issues of existence and uniqueness in minimization problems, and investigates properties of minimizing sequences. This chapter is devoted to a fundamental convexity-preserving operation for functions: the infimal convolution. Its properties are investigated, with special emphasis on the Moreau envelope, which is obtained by convolving a function with the halved squared norm.

In classical analysis, functional transforms make it possible to investigate problems from a different perspective and sometimes simplify their analysis. In convex analysis, the most suitable notion of a transform is the Legendre transform, which maps a function to its Fenchel conjugate. This transform is studied in detail in this chapter. In particular, it is shown that the conjugate of an infimal convolution is the sum of the conjugates.

The key result of this chapter is the Fenchel—Moreau theorem, which states that the proper convex lower semicontinuous functions are precisely those functions that coincide with their biconjugates. In this chapter, we exhibit several deeper results on conjugation. This powerful nonlinear principle extends the standard linear decomposition with respect to a closed linear subspace and its orthogonal complement.

Basic results concerning the proximal average and positively homogeneous functions are also presented.

### Theoryand Examples

Also discussed are the Moreau— Rockafellar theorem, which characterizes coercivity in terms of an interiority condition, and the Toland—Singer theorem, which provides an appealing formula for the conjugate of the difference. Of central importance in convex analysis are conditions guaranteeing that the conjugate of a sum is the infimal convolution of the conjugates. The main result in this direction is a theorem due to Attouch and Br9zis.

In turn, it gives rise to the Fenchel—Rockafellar duality framework for convex optimization problems. The subdifferential is a fundamental tool in the analysis of nondifferentiable convex functions. In this chapter we discuss the properties of subdifferentials and the interplay between the subdifferential and the Legendre transform.

In this chapter, we provide fundamental results on these relationships, as well as basic results on the steepest descent direction, the Chebyshev center, and the max formula that relates the directional derivative to the support function of the subdifferential at a given point. Further results concerning derivatives and subgradients are collected in this chapter.

Formulas for the subdifferential of a maximum and of an infimal convolution are provided, and the basic duality between differentiability and strict convexity is presented. Another highlight of this chapter is the Baillon—Haddad theorem, which states that nonexpansiveness and firm nonexpansiveness are identical properties for gradients of convex functions.

Finally, the subdifferential operator of the distance to a convex set is analyzed in detail. A convex optimization problem can be paired with a dual problem involving the conjugates of the functions appearing it its primal formulation. In this chapter, we study the interplay between primal and dual problems in the context of Fenchel—Rockafellar duality and, more generally, for bivariate functions.

## Convex Analysis and Optimization (CPSC F/MATH )

The latter approach leads naturally to saddle points and Lagrangians. Special attention is given to minimization under equality constraints and under inequality constraints. We start with a discussion of instances in which all primal solutions can be recovered from an arbitrary dual solution.

The theory of monotone set-valued operators plays a central role in many areas of nonlinear analysis. A prominent example of a monotone operator is the subdifferential operator investigated in Chapter Single-valued monotone operators will be seen to be closely related to the nonexpansive operators studied in Chapter 4. We are always looking for ways to improve customer experience on Elsevier. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website.

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