We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U i...

We contrast three decision rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: Γ-Maximin, Maximality, and E-admissibility. The rules extend Expected Utility theory as they require that an option is inadmissible if there is another that carries greater expected utility for each probability in a (closed) convex set. If the convex set i...

the notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $cat(0)$ space, where the curvature is bounded from above by zero. in fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. in this paper, w...

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...

This paper addresses robust deconvolution filtering when the system and noise dynamics are obtained by parametric system identification. Consistent with standard identificationmethods, the uncertainty in the estimated parameters is represented by an ellipsoidal uncertainty region. Three problems are considered: (1) computation of the worst case H2 performance of a given deconvolution filter in ...

‎We use of two notions functionally convex (briefly‎, ‎F--convex) and functionally closed (briefly‎, ‎F--closed) in functional analysis and obtain more results‎. ‎We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$‎, ‎then $bigcup_{alphain I}A_{alpha}$ is F--convex‎. ‎Moreover‎, ‎we introduce new definition o...

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result a...