A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization

Abstract In a recent issue of this Monthly, Mordukhovich, Nam, and Salinas pose and solve an interesting non-differentiable generalization of the Heron problem in the framework of modern convex analysis. In the generalized Heron problem, we are given k + 1 closed convex sets in ℝd equipped with its Euclidean norm and asked to find the point in the last set such that the sum of the distances to the first k sets is minimal. In later work, the authors generalize the Heron problem even further, relax its convexity assumptions, study its theoretical properties, and pursue subgradient algorithms for solving the convex case. Here, we revisit the original problem solely from the numerical perspective. By exploiting the majorizationminimization (MM) principle of computational statistics and rudimentary techniques from differential calculus, we are able to construct a very fast algorithm for solving the Euclidean version of the generalized Heron problem.

[1]  Frank Plastria,et al.  On the point for which the sum of the distances to n given points is minimum , 2009, Ann. Oper. Res..

[2]  Cun-Hui Zhang,et al.  A modified Weiszfeld algorithm for the Fermat-Weber location problem , 2001, Math. Program..

[3]  G. Hardy,et al.  What is Mathematics? , 1942, Nature.

[4]  Yoram Singer,et al.  Efficient projections onto the l1-ball for learning in high dimensions , 2008, ICML '08.

[5]  Leon Hirsch,et al.  Fundamentals Of Convex Analysis , 2016 .

[6]  Alfredo Marín,et al.  On the convergence of the Weiszfeld algorithm , 2002, Math. Program..

[7]  Kenneth Lange,et al.  Numerical analysis for statisticians , 1999 .

[8]  J. Borwein,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[9]  C. Michelot.  A finite algorithm for finding the projection of a point onto the canonical simplex of ∝n , 1986 .

[10]  Robert R. Meyer,et al.  Sufficient Conditions for the Convergence of Monotonic Mathematical Programming Algorithms , 1976, J. Comput. Syst. Sci..

[11]  Panos M. Pardalos,et al.  Convex optimization theory , 2010, Optim. Methods Softw..

[12]  Shay Gueron.  Two Applications of the Generalized Ptolemy Theorem , 2002, Am. Math. Mon..

[13]  Ying Xiong.  Nonlinear Optimization , 2014 .

[14]  K. Lange,et al.  EM algorithms without missing data , 1997, Statistical methods in medical research.

[15]  Boris S. Mordukhovich,et al.  Solving a Generalized Heron Problem by Means of Convex Analysis , 2012, Am. Math. Mon..

[16]  Arie Tamir,et al.  Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem , 1989, Math. Program..

[17]  A. Fiacco.  A Finite Algorithm for Finding the Projection of a Point onto the Canonical Simplex of R " , 2009 .

[18]  B. Mordukhovich,et al.  Applications of variational analysis to a generalized Heron problem , 2011, 1106.0088.

[19]  Boris S. Mordukhovich,et al.  Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem , 2011, J. Optim. Theory Appl..

[20]  Jack Brimberg,et al.  The Fermat—Weber location problem revisited , 1995, Math. Program..

[21]  Xiao-Li Meng,et al.  [Optimization Transfer Using Surrogate Objective Functions]: Discussion , 2000 .

[22]  Graham K. Rand,et al.  Facilities Location: Models and Methods , 1989 .

[23]  Jakob Krarup,et al.  On Torricelli's geometrical solution to a problem of Fermat , 1997 .

[24]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[25]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[26]  M. Teboulle,et al.  A smoothing technique for nondifferentiable optimization problems , 1988 .

[27]  Shay Gueron,et al.  The Fermat-Steiner Problem , 2002, Am. Math. Mon..

[28]  J. Brimberg.  FURTHER NOTES ON CONVERGENCE OF THE WEISZFELD ALGORITHM , 2003 .