Closed convex function

Abstract: Let S denote the functions that are analytic and univalent in the open unit disk and satisfy /(0) = 0 and/'(0) = 1 Also, let K, St, SR, and C be the subfamilies of 5 consisting of convex, starlike, real, and close-to-convex mappings, respectively The closed convex hull of each of these four families is determined as well as the extreme points for each Moreover, integral formulas are obtained for each hull in terms of the probability measures over suitable sets The extreme points for each family are particularly simple; for example, the Koebe functions f(z) = z/(l-xz)2, \x\ = 1, are the extreme points of cl co St These results are applied to discuss linear extremal problems over each of the four families A typical result is the following: Let J be a "nontrivial" continuous linear functional on the functions analytic in the unit disk The only functions in St that satisfy Re/(/) = max {ReJ(g) : geSt} are Koebe functions and there are only a finite number of them Introduction We shall be concerned with the closed convex hulls of various families of functions that are analytic and univalent in the open unit disk A={z eC : \z\< 1} For each family considered we obtain integral representations for the closed convex hull, and we determine all the extreme points In each case the extreme points are strikingly simple and familiar functions Thus we obtain a powerful tool for solving linear extremum problems over such families Let us establish some notation and outline our main results We shall let A denote the set of all functions analytic in A With the natural topology of uniform convergence on compact subsets of A, A is a locally convex linear topological space (15, p 150) Let S be the subset of A consisting of functions / that are univalent (one-to-one) in A and satisfy/(0)=0,/'(0) = l It is well known (7, p 217) that S is compact in A, or, equivalently, that S is closed and locally uniformly bounded (On each compact subset of A there is a common bound for all the functions in S) We shall be particularly interested in the following subfamilies of S K={fe S : /(A) is convex}, St={fe S : /(A) is starlike with respect to 0},

Abstract: Let $\Phi_0:\mathbb{R}^n\to \mathbb{R}\cup \{+\infty\}$ be a closed convex function and $\Phi_1:\mathbb{R}^n\to \mathbb{R}$ be a finite convex function that are bounded from below. Our goal is to build an algorithm which first minimizes the map $\Phi_0$ and secondly the map $\Phi_1$ over the set $S_0:= {\rm argmin}\, \Phi_0$. For that purpose, we define the following proximal-type algorithm: $$ -(x_{n+1}-x_n)/{\lambda_n}\in \partial_{\eta_n} (\Phi_0+\varepsilon_n \Phi_1) (x_{n+1}),\leqno({\mathcal A}_1) $$ where $(\lambda_n)$ is a positive step sequence, $(\eta_n)$ is a summable error sequence, and $(\varepsilon_n)$ is a control sequence tending toward $0$; $\partial_\eta$ denotes the $\eta$-approximate subdifferential. When $(\varepsilon_n)$ is a slow control, i.e., \,$\sum_{n=0}^{+\infty}=\varepsilon_n +\infty$, we prove that, under adequate conditions, the sequence $(x_n)$ defined by $({\mathcal A}_1)$ tends toward an element of $S_1:={\rm argmin\,}_{S_0}\Phi_1$. More generally, given finite convex functions $\Phi_2,\ldots,\Phi_N:\mathbb{R}^n\to \mathbb{R}$, let us define the sets $(S_i)_{i\in\{1,\ldots,N\}}$ by the recursive relation $S_i:={\rm argmin\,}_{S_{i-1}}\Phi_i$. We introduce an extension of algorithm $({\mathcal A}_1)$ to minimize hierarchically each function $\Phi_i$ on the set $S_{i-1}$, for $i\in\{1,\ldots,N\}$.

Abstract: In a previous paper [3] we solved the Fekete-Szeg6 problem of maximizing l a 3 - 2 a2l, 2 ~ [0, 1], for close-to-convex functions. The largest number 20 for which [a a - 20 a2[ is maximized by the Koebe function z/(1 - z) 2 is 20 = 1/3. Now we generalize this result to C (fl), fl > 1, showing that the largest number 20 (fl) for which l a 3 - 20 (fl)a2[ is maximized over C (fl) by k a with

Abstract: For a symmetric positive semidefinite linear system of equations $$\mathcal{Q}{{\varvec{x}}}= {{\varvec{b}}}$$ , where $${{\varvec{x}}}= (x_1,\ldots ,x_s)$$ is partitioned into s blocks, with $$s \ge 2$$ , we show that each cycle of the classical block symmetric Gauss–Seidel (sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form $$\frac{1}{2}\Vert {{\varvec{x}}}-{{\varvec{x}}}^k\Vert _\mathcal{T}^2$$ , where $$\mathcal{T}$$ is a symmetric positive semidefinite matrix related to the sGS decomposition of $$\mathcal{Q}$$ and $${{\varvec{x}}}^k$$ is the previous iterate. By leveraging on such a connection to optimization, we are able to extend the result (which we name as the block sGS decomposition theorem) for solving convex composite QP (CCQP) with an additional possibly nonsmooth term in $$x_1$$ , i.e., $$\min \{ p(x_1) + \frac{1}{2}\langle {{\varvec{x}}},\,\mathcal{Q}{{\varvec{x}}}\rangle -\langle {{\varvec{b}}},\,{{\varvec{x}}}\rangle \}$$ , where $$p(\cdot )$$ is a proper closed convex function. Based on the block sGS decomposition theorem, we extend the classical block sGS method to solve CCQP. In addition, our extended block sGS method has the flexibility of allowing for inexact computation in each step of the block sGS cycle. At the same time, we can also accelerate the inexact block sGS method to achieve an iteration complexity of $$O(1/k^2)$$ after performing k cycles. As a fundamental building block, the block sGS decomposition theorem has played a key role in various recently developed algorithms such as the inexact semiproximal ALM/ADMM for linearly constrained multi-block convex composite conic programming (CCCP), and the accelerated block coordinate descent method for multi-block CCCP.