A SIMPLE DERIVATION OF SUFFICIENT CONDITIONS FOR A LOCAL MINIMUM OF A LIPSCHITZIAN FUNCTION

1989 ◽  
Vol 22 (1) ◽  
Author(s):  
Marcin Studniarski
Keyword(s):  
Local Minimum ◽  
Sufficient Conditions ◽  
Simple Derivation ◽  
Lipschitzian Function

scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization

2009 ◽  
Vol 61 (1) ◽  
pp. 205-221 ◽  
Author(s):  
M. Marshall
Keyword(s):  
Global Optimization ◽  
Local Minimum ◽  
Final Section ◽  
Sufficient Conditions ◽  
Minimum Point ◽  
Sums Of Squares ◽  
Compact Set ◽  
Theoretical Interest ◽  
Feasible Set ◽  
Degree Bounds

Abstract. Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

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