Finite Dimensional Convexity and Optimization Book

	Notation
1	Convexity in R[superscript n]	1
1.1	Basic concepts	1
1.2	Topological properties of convex sets	8
2	Separation and Polarity	21
2.1	Separation of convex sets	21
2.2	Polars of convex sets and orthogonal subspaces	28
2.3	Application to Minkowski-Farkas' lemma	29
3	Extremal Structure of Convex Sets	37
3.1	Extreme points and faces of convex sets	37
3.2	Application to linear inequalities. Weyl's theorem	40
3.3	Extreme points and extremal subsets of a polyhedral convex set	45
4	Linear Programming	51
4.1	Necessary and sufficient conditions of optimality	52
4.2	The duality theorem of linear programming	54
4.3	The simplex method	60
5	Convex Functions	73
5.1	Basic definitions and properties	73
5.2	Continuity theorems	76
5.3	Continuity properties of collections of convex functions	79
6	Differential Theory of Convex Functions	87
6.1	The Hahn-Banach dominated extension theorem	88
6.2	Sublinear functions	89
6.3	Support functions	91
6.4	Directional derivatives	92
6.5	Subgradients and subdifferential of a convex function	93
6.6	Differentiability of convex functions	98
6.7	Differential continuity for convex functions	100
7	Convex Optimization With Convex Constraints	109
7.1	The minimum of a convex function f: R[superscript n] [approaches]R	110
7.2	Kuhn-Tucker Conditions	111
7.3	Value functions	123
8	Non Convex Optimization	129
8.1	Quasi-convex functions	129
8.2	Minimization of quasi-convex functions	134
8.3	Differentiable optimization	137
A	Appendix	141
	Bibliography	147
	Index	151