A practical approach to optimization

We present a new approach for finding a minimal value of an arbitrary function assuming only its continuity. The process avoids verifying Lagrange- or KKT-conditions. The method enables us to obtain a Brouwer fixed point (of a continuous function mapping from a cube into itself).

[1]  Nicholas I. M. Gould,et al.  A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds , 1997, Math. Comput..

[2]  Jorge Nocedal,et al.  A trust region method based on interior point techniques for nonlinear programming , 2000, Math. Program..

[3]  L. E. J. Brouwer,et al.  Über Jordansche Mannigfaltigkeiten , 1911 .

[4]  Poom Kumam,et al.  An Elementary Proof of the Brouwer Fixed Point Theorem , 2019 .

[5]  Riccardo Poli,et al.  Particle swarm optimization , 1995, Swarm Intelligence.

[6]  Jorge Nocedal,et al.  An Interior Point Algorithm for Large-Scale Nonlinear Programming , 1999, SIAM J. Optim..

[7]  Poom Kumam,et al.  A Graphical Proof of the Brouwer Fixed Point Theorem , 2015 .

[8]  Carlos A. Coello Coello,et al.  Constraint-handling in nature-inspired numerical optimization: Past, present and future , 2011, Swarm Evol. Comput..

[9]  J. Dennis,et al.  Pattern search algorithms for mixed variable general constrained optimization problems , 2003 .

[10]  Sompong Dhompongsa,et al.  A Simple Proof of the Brouwer Fixed Point Theorem , 2015 .

[11]  Charles Audet,et al.  Analysis of Generalized Pattern Searches , 2000, SIAM J. Optim..

[12]  E. Zeidler,et al.  Fixed-point theorems , 1986 .

[13]  M. E. H. Pedersen.  Good Parameters for Particle Swarm Optimization , 2010 .

[14]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..