Study On Monotonization Convexification And Concavification Methods For Some Mathematical Programming

Our time is witnessing the rapid growth of a new field, global optimization.Many new theoretical , algorithmic, and computational contributions of globaloptimization have been used to solve many problems in science and engineering. Sincethere exist multiple local optima that differ from the global solution, and the traditionalminimization techniques for nonlinear programming are devised for obtaining localoptimal solutions, they can not be used smoothly to solve the global optimizationproblems. During the past several decades, great development has been obtained in thetheoretical and algorithmic aspects of global optimization due to the importantpractical applications. These developed approaches mainly consist of two categories:deterministric approaches and stochastic approaches.In this paper, we will more deeply research on the base of the works. This paperis organized as follows:The first part, a brief introduction is given to the existing main classes ofdeterministric global optimization approaches.The second part , we give the preparation knowledge to the whole paper.The third part, firstly, we propose new convexification and concavificationtransformations to convert strictly monotone function into a convex or concavefunction. Then , we prove that the original programming problem can be convertedinto an equivalent concave minimization problem, or reverse convex programmingproblem or canonical D.C. programming problem. Then the global optimal solution ofthe original problem can be obtained by solving the converted concave minimizationproblem, or reverse convex programming problem or canonical D.C. programmingproblem using the existing algorithms about them.The 4-th part, in section 4.2, we deals with some non-monotonic programmingproblems with monotonic constraints . Firstly, a nonlinear programming problem withmultiple constraints can be converted into a programming problem with singleconstraint via maximum entropy function. Secondly, we give a monotonictransformation for the converted single programming problem and prove theequivalence between the converted monotone programming problem and theprogramming problem with single constraint. [7] gave a monotonizationtransformation, but it didn't prove the equivalence of monotonization transformation,in section 4.3, we will prove the equivalence of monotonization transformation. Insection 4.4, we propose several convexification and concavification transformations toconvert a non-convex and non-concave objective function into a convex or concave inthe programming problems with convex or concave constraint functions.The 5-th part, we summarize the total paper, and put forward some problemwhich should be solved in perhaps development direction in the future.In the paper, main results gather in the third part, the 4-th part .