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David H. Gutman(dgutmanandrew.cmu.edu) Abstract: The condition number of a smooth convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex function is precisely the square of the diametertowidth ratio of a canonical ellipsoid associated to the function. Furthermore, the condition number of a function bounds the linear rate of convergence of the gradient descent algorithm for unconstrained minimization. We propose a condition number of a smooth convex function relative to a reference polytope. This relative condition number is defined as the ratio of a relative smooth constant to a relative strong convexity constant of the function, where both constants are relative to the reference polytope. The relative condition number extends the main properties of the traditional condition number. In particular, we show that the condition number of a quadratic convex function relative to a polytope is precisely the square of the diametertofacialdistance ratio of a scaled polytope for a canonical scaling induced by the function. Furthermore, we illustrate how the relative condition number of a function bounds the linear rate of convergence of firstorder methods for minimization of the function over the polytope. Keywords: Condition, facial distance, linear convergence Category 1: Convex and Nonsmooth Optimization Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Technical Report. Carnegie Mellon University Download: [PDF] Entry Submitted: 02/02/2018 Modify/Update this entry  
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