On the Triality Theory in Global Optimization

TL;DR

This paper proves the triality theory for unconstrained global optimization problems, clarifying conditions under which various duality principles hold, and introduces a new weak saddle min-max duality theorem, solving an open problem from 2003.

Contribution

It provides a rigorous proof of the triality theory for general unconstrained problems and introduces a new weak saddle duality theorem, resolving a longstanding open problem.

Findings

Triality theory holds strongly when primal and dual have same dimension.

Double-min duality holds weakly in certain cases.

A new weak saddle min-max duality theorem is established.

Abstract

Triality theory is proved for a general unconstrained global optimization problem. The method adopted is simple but mathematically rigorous. Results show that if the primal problem and its canonical dual have the same dimension, the triality theory holds strongly in the tri-duality form as it was originally proposed. Otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a super-symmetrical form as it was expected. Additionally, a complementary weak saddle min-max duality theorem is discovered. Therefore, an open problem on this statement left in 2003 is solved completely. This theory can be used to identify not only the global minimum, but also the largest local minimum, maximum, and saddle points. Application is illustrated. Some fundamental concepts in optimization and remaining challenging problems in canonical duality theory are discussed.