| In this paper, separable convex optimization problems with equality or inequality linear constraints are analyzed in the image space from the geometric point of view. In general, a separate function is needed to separate two sets, K and H, in the image space of constrained extremum problems when their optimal conditions are analyzed in the image space approach. At present, many scholars and experts devote themselves to construct specific linear or abstract nonlinear separate functions. Abstract nonlinear separate functions do not need convexity assumptions while separating two sets in the image space, but the drawback is that their function expressions do not be known, which is less useful to solve the practical problems. In this paper, the separable convex optimization problems with equality or inequality linear constraints are described uniformly and some equivalence conclusions related are proved. A specific nonlinear weak separate function is constructed based on a nonlinear scalarization function (â–³-function). According to its generalized saddle point of Lagrangian function, some optimal conditions are obtained. |
