Geometry Property Of Conic Linear Optimization In Reflexive Banach Spaces

Now, there is a hot discussion on conic optimization, the study is mainly on the basic theory of conic optimization, in order to find the algorithms of conic optimization. Also many Chinese scholars begin to study on conic optimization.This is a survey of obtained results of the basic theory of conic optimization. We introduce the basic concepts, the basic properties, and other relative basic theory. Meanwhile we introduce some relative theory of Applied Functional Analysis. The properties of conic optimization may be different for the dimension of the space. We mainly present the basic theory of conic optimization in infinite dimension spaces, provide the possibility of find the right algorithms.We consider the following conic optimization:If the above optimization is in finite dimension spaces, it can be written as the following:Let X be the set of k-dimensional symmetry matrix, define Lowner order"≥"on X, namely the sufficient and necessary condition of x≥w is x -w is semi-positive definite. If K ={0} and C = { x∈Xx≥0}, then(GP)is semi-positive definite programming. If C is a closed set in X ,we can transform the optimization into conic optimization by using the closed convex cone which generated by C×{1}, the details are in reference [4] and [19];also can use other method to transform the optimization into conic optimization, the details are in reference [9].Conic optimization origins from economics, so scholars firstly focus on the conic optimization in finite dimensional spaces. Vera [1 2 ] considered the existence of linear programming with asymptotic parameters, it plays an important part in understanding the ill-posedness of optimization and asymptotic parameter problems. Later, Freund and Vera study the complexity of using the ellipse algorithm to request the∈-optimal solution, in 1999.For the dual optimization of the dual optimization is primal optimization in reflexive Banach Spaces and the properties of the special spaces provide the possibility of considering the geometry properties of conic optimization, recently many scholars focus on this subject. We mainly introduce the geometry properties of conic optimization, in reflexive Banach Spaces.We consider the following conic optimization(GP) and its dual optimization(GD) in reflexive Banach Spaces: and When (X , ||.||) and (Y , ||.||) is reflexive Banach Spaces,the correspond conjugate spaces are ( X ~*, ||.||~*) and (Y ~*, ||.||~*),C is an closed convex cone in X , K is an closed convex cone in Y , b∈Y,α∈X~*, A:X→Yis a continuous bounded operator, namely A is a linear continuous operator, A ~* : Y~*→X~* is the conjugate operator of A .Especially,when K ={0}, denote (GP) and (GD) by (P) and (D), namely: andIn chaper 1, we introduce the development of optimization and algorithm.In chaper 2, we introduce the basic concepts and theory used in this paper, present the virtue of reflexive Banach Spaces, and the property of norm (lemma 2.1).Lemma 2.1 For any x∈X,there exist x ~*∈X~*satisfied || x ~* ||~*=1 and || x ||= (x~* ,x).We introduce the proof of the above lemma, it can be proofed by the definition of subdifferential. The property can be considered as a corollary of Hahn-Banach theorem. Freund and Vera gave the proof in finite dimension linear normed space in reference [25].In Chapter 3, we introduce a sufficient condition for the duality of conic optimization. Firstly, we introduce the weak dual theorem and its proof.Theorem 3.1 (weak-dual theorem) If (GP) and (GD) is feasible , then z ~*≥v~*.Theorem 3.2 Assume z ~* is finite, and intC ~*≠φ. If there exists∈>0, satisfied the following condition: is bounded, then the optimal value of (GP) can be obtained, meanwhile z ~* = v~*.In Chapter 4 ,we introduce some geometry property of the level sets of the objective function of conic optimization (P) and (D).Assumption ( A) : z ~* is finite, and intC ~*≠θ.We introduce the property of inner measure: Theorem 4.1 Let K be a convex cone in X . If inner measure of K ~* can be obtained at S~0∈int k~*, |S~0|_~*=1, then the following condition holds. (i)for any (ii)ifThen we introduce some geometry property of the level sets of the objective function of conic optimization (P) and (D):Theorem 4.2 Suppose that assumption (A) holds. If 0 < Rε<+∞, then z ~* = v~*, and If R_ε=0,then z~~* = v~~*,and r_δ=+∝.At last, we introduce some geometry property of the level sets of the objective function of conic optimization (GP) and (GD):Theorem 4.3 Assume z ~* is finite,and int C ~*≠φ,if 0 < R_ε<+∞, then z ~* = v~*,and...