We study the online convex covering problem and online convex packing problem. The (offline) convex covering problem is modeled by the following convex program: min ~ x∈Rn+ f(~x) s.t. A~x ≥ ~1 where f : R+ 7→ R+ is a monotone and convex cost function, and A is an m × n matrix with non-negative entries. Each row of the constraint matrix A corresponds to a covering constraint. In the online probl...

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their ‘concavity index’, m. Such polygons are called m-convex polygons and are characterized by having up to m indentations in their perimeter. We first describe how we conjectured the (isotropic) generating function for the case m = 2 using a numerical procedure ba...

Let Z andX be Banach spaces, U ⊂ Z be an open convex set and f : U → X be a mapping. We say that f is a d.c. mapping if there exists a continuous convex function h on U such that y ◦ f + h is a continuous convex function for each y ∈ Y , ‖y∗‖ = 1. We say that f : U → X is locally d.c. if for each x ∈ U there exists open convex U ′ such that x ∈ U ′ ⊂ U and f |U ′ is d.c. This notion of d.c. map...

For some applications it is desired to approximate a set of m data points in IR with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n+ 1)(n+ 2)/2 parameters, s ≤ m, so as to minimize the approximation error...

In this paper, a new inequality for generalized convex functions which is related to the left side of generalized Hermite-Hadamard type inequality is obtained. Some applications for some generalized special means are also given.