Introduction to Unconstrained Optimization - Scilab

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-10 -8 -6 -4 -2 0 2 4 6 8 10
Figure 26: Contour of a quadratic function – The linear system associated with a
zero gradient is incompatible.
2.7 Notes and references
The section 2, which focus on convex functions is inspired from ”Convex Optimization”by
Boyd and Vandenberghe [5], chapter 2, ”Convex sets”and chapter 3, ”Convex
functions”. A complementary and vivid perspective is given by Stephen Boyd in a
series of lectures available in videos on the website of Stanford’s University.
Most of this section can also be found in ”Convex Analysis and Optimization”
by Bertsekas [3], chapter 1, ”Basic Convexity Concepts”.
Some of the examples considered in the section 2.5 are presented in [5], section
3.1.5 ”Examples”. The nonconvex function f(x 1 , x 2 ) = x 1 /(1 + x 2 2) presented in section
2.5 is presented by Stephen Boyd in his Stanford lecture ”Convex Optimization
I”, Lecture 5.
The quadratic functions from section 2.6 are presented more briefly in Gill, Murray
and Wright [6].
The book by Luenberger [9] presents convex sets in Appendix B, ”Convex sets”
and presents convex functions in section 6.4, ”Convex and concave functions”.
2.8 Exercises
Exercise 2.1 (Convex hull - 1 ) Prove that a set is convex if and only if it contains every convex
combinations of its points.
Exercise 2.2 (Convex hull - 2 ) Prove that the convex hull is the smallest convex set that
contains C.
Exercise 2.3 (Convex function - 1 ) Prove that the sum of two convex functions is a convex
function.
Exercise 2.4 (Convex function - 2 ) Prove that the level sets of a convex function are convex.
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