v2010.10.26 - Convex Optimization

5.4. EDM DEFINITION 411is the fact: this earliest instance of duality is a two-dimensional Euclideandistance geometry problem known as a Fermat point (Figure 118) namedafter the French mathematician. Given N distinct points in the plane{x i ∈ R 2 , i=1... N} , the Fermat point y is an optimal solution tominimizeyN∑‖y − x i ‖ (924)i=1a convex minimization of total distance. The historically first dual problemformulation asks for the smallest equilateral triangle encompassing (N = 3)three points x i . Another problem dual to (924) (Kuhn 1967)maximize{z i }subject toN∑〈z i , x i 〉i=1N∑z i = 0i=1‖z i ‖ ≤ 1∀i(925)has interpretation as minimization of work required to balance potentialenergy in an N-way tug-of-war between equally matched opponents situatedat {x i }. [374]It is not so straightforward to write the Fermat point problem (924)equivalently in terms of a Gram matrix from this section. Squaring insteadminimizeαN∑‖α−x i ‖ 2 ≡i=1minimizeD∈S N+1 〈−V , D〉subject to 〈D , e i e T j + e j e T i 〉 1 2 = ďij∀(i,j)∈ I−V DV ≽ 0 (926)yields an inequivalent convex geometric centering problem whoseequality constraints comprise EDM D main-diagonal zeros and knowndistances-square. 5.10 Going the other way, a problem dual to totaldistance-square maximization (Example 6.7.0.0.1) is a penultimate minimumeigenvalue problem having application to PageRank calculation by searchengines [231,4]. [340]Fermat function (924) is empirically compared with (926) in [61,8.7.3],but for multiple unknowns in R 2 , where propensity of (924) for producing5.10 α ⋆ is geometric center of points x i (973). For three points, I = {1,2,3} ; optimalaffine dimension (5.7) must be 2 because a third dimension can only increase totaldistance. Minimization of 〈−V,D〉 is a heuristic for rank minimization. (7.2.2)