Chasing Convex Bodies and Functions
نویسندگان
چکیده
We consider three related online problems: Online Convex Optimization, Convex Body Chasing, and Lazy Convex Body Chasing. In Online Convex Optimization the input is an online sequence of convex functions over some Euclidean space. In response to a function, the online algorithm can move to any destination point in the Euclidean space. The cost is the total distance moved plus the sum of the function costs at the destination points. Lazy Convex Body Chasing is a special case of Online Convex Optimization where the function is zero in some convex region, and grows linearly with the distance from this region. And Convex Body Chasing is a special case of Lazy Convex Body Chasing where the destination point has to be in the convex region. We show that these problems are equivalent in the sense that if any of these problems have an O(1)-competitive algorithm then all of the problems have an O(1)competitive algorithm. By leveraging these results we then obtain the first O(1)-competitive algorithm for Online Convex Optimization in two dimensions, and give the first O(1)-competitive algorithm for chasing linear subspaces. We also give a simple algorithm and O(1)-competitiveness analysis for chasing lines.
منابع مشابه
Nested Convex Bodies are Chaseable
In the Convex Body Chasing problem, we are given an initial point v0 ∈ R and an online sequence of n convex bodies F1, . . . , Fn. When we receive Fi, we are required to move inside Fi. Our goal is to minimize the total distance travelled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an Ω( √ d) lower bound on the competitive ratio, and conject...
متن کاملHermite-Hadamard inequalities for $mathbb{B}$-convex and $mathbb{B}^{-1}$-convex functions
Hermite-Hadamard inequality is one of the fundamental applications of convex functions in Theory of Inequality. In this paper, Hermite-Hadamard inequalities for $mathbb{B}$-convex and $mathbb{B}^{-1}$-convex functions are proven.
متن کاملBernstein's polynomials for convex functions and related results
In this paper we establish several polynomials similar to Bernstein's polynomials and several refinements of Hermite-Hadamard inequality for convex functions.
متن کامل(m1,m2)-AG-Convex Functions and Some New Inequalities
In this manuscript, we introduce concepts of (m1,m2)-logarithmically convex (AG-convex) functions and establish some Hermite-Hadamard type inequalities of these classes of functions.
متن کامل