Convex and concave functions

Convexity or concavity of functions is an important property in numerical optimization. 

Conversely, a function f x) is strictly concave if —f x) is strictly convex. 

A global optimum can be easily obtained, if the following conditions apply. 

This section presents (i) the definitions and properties of convex and concave functions, (ii) the definitions of continuity, semicontinuity and subgradients, (iii) the definitions and properties of differentiable convex and concave functions, and (iv) the definitions and properties of local and global extremum points. 

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This chapter discusses the elements of convex analysis which are very important in the study of optimization problems. In section 2.1 the fundamentals of convex sets are discussed. In section 2.2 the subject of convex and concave functions is presented, while in section 2.3 generalizations of convex and concave functions are outlined. 

Illustration 2.2.2 Figure 2.8 shows the epigraph and hypograph of a convex and concave function. Theorem 2.2.1... 

Remark 3 Convex and concave functions may not be continuous everywhere but the points of discontinuity have to be on the boundary of S. 

Illustration 2.2.5 Figure 2.11 shows a differentiable convex and concave function, as well as their linearizations around a point x1. Note that the linearization always underestimates the convex function and always overestimates the concave function. 

Remark 2 Convex and concave functions do not satisfy properties (ii) and (iii) of the quasi-convex and quasi-concave functions. For instance, it is true that the reciprocal of a positive concave function is convex, but the reverse does not hold. As an example consider the function f(x) = ex which is convex and whose reciprocal is also convex. 

Section 2.3 focuses on the generalizations of convex and concave functions and treats the quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions, and their properties. Further reading in this subject is the excellent book of Avriel et al. (1988). 

Part 1, comprised of three chapters, focuses on the fundamentals of convex analysis and nonlinear optimization. Chapter 2 discusses the key elements of convex analysis (i.e., convex sets, convex and concave functions, and generalizations of convex and concave functions), which are very important in the study of nonlinear optimization problems. Chapter 3 presents the first and second order optimality conditions for unconstrained and constrained nonlinear optimization. Chapter 4 introduces the basics of duality theory (i.e., the primal problem, the perturbation function, and the dual problem) and presents the weak and strong duality theorem along with the duality gap. Part 1 outlines the basic notions of nonlinear optimization and prepares the reader for Part 2. 

To. .generalize the results of Examples 8.2 and 83, we define convex and concave functions. As presented in the introductory calculus course, the simplest version pertains to functions having at least two derivatives. In that case, a function is convex (concave upward) if its second derivative is everywhere greater than or equal to zero. A function is concave (concave downward) if its second derivative is everywhere less than or equal to zero, as shown in Figure 8.24... 

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