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The paper introduces several new concepts for solving nonconvex or nonsmoothoptimization problems, including convertible nonconvex function, exactconvertible nonconvex function and differentiable convertible nonconvexfunction. It is proved herein many nonconvex functions or nonsmooth (ordiscontinuous) functions are actually convertible nonconvex functions andconvertible nonconvex function operations such as addition, subtraction,multiplication or division result in convertible nonconvex functions. Thesufficient condition for judging a global optimal solution to unconstrainedoptimization problems with differentiable convertible nonconvex functions isproved, which is equivalent to Karush-Kuhn-Tucker(KKT) condition. Two Lagrangefunctions of differentiable convertible nonconvex function are defined withtheir dual problems defined accordingly. The strong duality theorem is proved,showing that the optimal objective value of the global optimal solution isequal to the optimal objective value of the dual problem, which is equivalentto KKT condition. An augmented Lagrangian penalty function algorithm isproposed and its convergence is proved. So the paper provides a new idea forsolving unconstrained nonconvex or non-smooth optimization problems and avoidssubdifferentiation or smoothing techniques by using some gradient searchalgorithms, such as gradient descent algorithm, Newton algorithm and so on.

Optimization Condition and Algorithm of Optimization with Convertible Nonconvex Function