An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence

Semi-infinite programming can be used to model a large variety of complex optimization problems. The simple description of such problems comes at a price: semi-infinite problems are often harder to solve than finite nonlinear problems. In this paper we combine a classical adaptive discretization method developed by Blankenship and Falk and techniques regarding a semi-infinite optimization problem as a bi-level optimization problem. We develop a new adaptive discretization method which combines the advantages of both techniques and exhibits a quadratic rate of convergence. We further show that a limit of the iterates is a stationary point, if the iterates are stationary points of the approximate problems

A generalized projection-based scheme for solving convex constrained optimization problems

In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility problems by iteratively constraining the objective function from above until the feasibility problem is inconsistent. For each of the feasibility problems one may apply any of the existing projection methods for solving it. In particular, the scheme allows the use of subgradient projections and does not require exact projections onto the constraints sets as in existing similar methods. We also apply the newly introduced concept of superiorization to optimization formulation and compare its performance to our scheme. We provide some numerical results for convex quadratic test problems as well as for real-life optimization problems coming from medical treatment planning.Comment: Accepted to publication in Computational Optimization and Application

On the Variance of Additive Random Variables on Stochastic Polyhedra

Let $a_i i:= 1,\dots,m.$ be an i.i.d. sequence taking values in $\mathbb{R}^n$. Whose convex hull is interpreted as a stochastic polyhedron $P$. For a special class of random variables which decompose additively relative to their boundary simplices, eg. the volume of $P$, integral representations of their first two moments are given which lead to asymptotic estimations of variances for special "additive variables" known from stochastic approximation theory in case of rotationally symmetric distributions

An improved asymptotic analysis of the expected number of pivot steps required by the simplex algorithm

Let $a_1,\dots,a_m$ be i.i .d. vectors uniform on the unit sphere in $\mathbb{R}^n$, $m\ge n\ge3$ and let $X$:= {$x \in \mathbb{R}^n \mid a ^T_i x\leq 1$} be the random polyhedron generated by. Furthermore, for linearly independent vectors $u$, $\bar u$ in $\mathbb{R}^n$, let $S_{u, \bar u}(X)$ be the number of shadow vertices of $X$ in $span (u, \bar u$). The paper provides an asymptotic expansion of the expectation value $E (S_{u, \bar u})$ for fixed $n$ and $m\to\infty$. The first terms of the expansion are given explicitly. Our investigation of $E (S_{u, \bar u})$ is closely connected to Borgwardt's probabilistic analysis of the shadow vertex algorithm - a parametric variant of the simplex algorithm. We obtain an improved asymptotic upper bound for the number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data

A Simple Integral Representation for the Second Moments of Additive Random Variables on Stochastic Polyhedra

Let $a_1, i:=1,\dots,m$, be an i.i.d. sequence taking values in $\mathbb{R}^n$, whose convex hull is interpreted as a stochastic polyhedron $P$. For a special class of random variables, which decompose additively relative to their boundary simplices, eg. the volume of $P$, simple integral representations of its first two moments are given in case of rotationally symmetric distributions in order to facilitate estimations of variances or to quantify large deviations from the mean

A comparison method for expectations of a class of continuous polytope functionals

Let $a_1,\dots,a_n$ be independent random points in $\mathbb{R}^d$ spherically symmetrically but not necessarily identically distributed. Let $X$ be the random polytope generated as the convex hull of $a_1,\dots,a_n$ and for any $k$-dimensional subspace $L\subseteq \mathbb{R}^d$ let $Vol_L(X) :=\lambda_k(L\cap X)$ be the volume of $X\cap L$ with respect to the $k$-dimensional Lebesgue measure $\lambda_k, k=1,\dots,d$. Furthermore, let $F^{(i)}$(t):= $\bf{Pr}$ \)(\(\Vert a_i \|_2\leq t), $t \in \mathbb{R}^+_0$ , be the radial distribution function of $a_i$. We prove that the expectation functional $\Phi_L$($F^{(1)}, F^{(2)},\dots, F^{(n)})$ := $E(Vol_L(X)$) is strictly decreasing in each argument, i.e. if $F^{(i)}(t) \le G^{(i)}(t)t$, $t \in {R}^+_0$, but $F^{(i)} \not\equiv G^{(i)}$, we show $\Phi$ $(\dots, F^{(i)}, \dots$) > $\Phi(\dots,G^{(i)},\dots$). The proof is clone in the more general framework of continuous and $f$- additive polytope functionals

On the Approximation of a Ball by Random Polytopes

Let ($a_i)_{i\in \bf{N}}$ be a sequence of identically and independently distributed random vectors drawn from the $d$-dimensional unit ball $B^d$and let $X_n$:= convhull $(a_1,\dots,a_n$) be the random polytope generated by $(a_1,\dots\,a_n)$. Furthermore, let $\Delta (X_n)$ : = (Vol $B^d$ \ $X_n$) be the deviation of the polytope's volume from the volume of the ball. For uniformly distributed $a_i$ and $d\ge2$, we prove that tbe limiting distribution of $\frac{\Delta (X_n)} {E(\Delta (X_n))}$ for $n\to\infty$ satisfies a 0-1-law. Especially, we provide precise information about the asymptotic behaviour of the variance of $\Delta (X_n$). We deliver analogous results for spherically symmetric distributions in $B^d$ with regularly varying tail

On the expected number of shadow vertices of the convex hull of random points

Let $a_1,\dots,a_m$ be independent random points in $\mathbb{R}^n$ that are independent and identically distributed spherically symmetrical in $\mathbb{R}^n$. Moreover, let $X$ be the random polytope generated as the convex hull of $a_1,\dots,a_m$ and let $L_k$ be an arbitrary $k$-dimensional subspace of $\mathbb{R}^n$ with $2\le k\le n-1$. Let $X_k$ be the orthogonal projection image of $X$ in $L_k$. We call those vertices of $X$, whose projection images in $L_k$ are vertices of $X_k$as well shadow vertices of $X$ with respect to the subspace $L_k$ . We derive a distribution independent sharp upper bound for the expected number of shadow vertices of $X$ in $L_k$