Constructing test functions for global optimization using continuous formulations of graph problems

Exact Penalty Functions for Nonlinear Integer Programming Problems

“…where z is a minimum point of (1) and x is a minimum point of (2). Recalling the first part of the proof, we have that, ∀ ε ∈ ]0,ε], the point x is also a minimum point of (1); hence, using Assumption (A2), we have…”

“…where W ⊆ X ⊂ R n , and ϕ(·, ε) : R n → R. In (1), (2) and in the sequel, min denotes the global minimum.…”

On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems

“…The main idea of another typical class of available approaches for nonlinear mixed discrete programming problems is to transform discrete formulations into continuous ones. Nowadays, continuous formulations of discrete problems have been widely developed in the literature, for example, [6,12,19,25,26,29,35,36] and references therein for relevant results on the equivalence between integer and continuous programming; [1,3,10,11,31,32,34] and references therein for some relevant applications where continuous formulations have been successfully used. The main idea of most literature is to design continuous formulations that enable optimization methods for continuous problems to be applied to certain types of discrete optimization problems.…”

New results on the equivalence between zero-one programming and continuous concave programming

“…Note that the employment of reformulations of a given (possibly mixed) integer problem in a continuous space can be a practicable technique for finding good solutions even when dealing with large-scale problems (see e.g., [1,2,8,9,13]). …”