In stochastic convex optimization the goal is to minimize a convex function F (x) . = Ef∼D[f(x)] over a convex set K ⊂ R where D is some unknown distribution and each f(·) in the support of D is convex over K. The optimization is commonly based on i.i.d. samples f, f, . . . , f from D. A standard approach to such problems is empirical risk minimization (ERM) that optimizes FS(x) . = 1 n ∑ i≤n f...

An important class of planar straight-line drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of te...

Abstract A method of Sequential Log-Convex Programming (SLCP) is constructed that exploits the log-convex structure present in many engineering design problems. The mathematical Geometric (GP) combined with ability Quadratic Program (SQP) to accommodate a wide range objective and constraint functions, resulting practical algorithm can be adopted little no modification existing practices. Three ...

One useful generalization of the convex hull of a set S of n points is the-strongly convex-hull. It is deened to be a convex polygon P with vertices taken from S such that no point in S lies farther than outside P and such that even if the vertices of P are perturbed by as much as , P remains convex. It was an open question 1 as to whether an-strongly convex O()-hull existed for all positive. W...

A sequence (xn)n 0 of positive real numbers is log-convex if the inequality xn xn−1xn+1 is valid for all n 1 . We show here how the problem of establishing the log-convexity of a given combinatorial sequence can be reduced to examining the ordinary convexity of related sequences. The new method is then used to prove that the sequence of Motzkin numbers is log-convex.

where λis are the eigenvalues of the Hessian D 2u. Namely, any global convex solution to (1.1) in R must be a quadratic polynomial. Recall the classical result, any global convex solution in R to the Laplace equation △u = λ1+ · · ·+λn = c or the Monge-Ampère equation log detD2u = log λ1+ · · ·+ log λn = c must be quadratic. Equation (1.1) originates from special Lagrangian geometry [HL]. The (L...

We propose a new parametric model for continuous data, a “g-model”, on the basis of gradient maps of convex functions. It is known that any multivariate probability density on the Euclidean space is uniquely transformed to any other density by using the gradient map of a convex function. Therefore the statistical modeling for quantitative data is equivalent to design of the gradient maps. The e...