Global optimality conditions in mathematical programming and optimal control.

摘要

We derive new first-order necessary and sufficient optimality conditions characterizing global minimizers in mathematical programming and optimal control problems. These conditions are based on level sets of an objective functional and they do not assume special structure of a problem (convexity, linearity, etc.). For a mathematical programming problem of minimization of a smooth functional on some compact convex set with equality nonlinear constraints, we derive first-order optimality conditions in the form of a generalized Lagrange multiplier rule. This rule should hold for any point from the level set of the objective functional corresponding to a global minimizer. We demonstrate that these necessary conditions become sufficient ones for optimality under additional assumption of non-degeneracy of the Lagrange multiplier rule.; We also study global optimality conditions for free time optimal control problem which includes the classical minimum-time problem. We derive necessary conditions for global optimality of relaxed controls in terms of Pontryagin minimum principle for any relaxed control from the level set of the objective functional. It is shown that these optimality conditions are sufficient for global optimality if the minimum principle is non-degenerated at least at one point on a time interval. In particular, we derive that if some relaxed control satisfies non-degenerated Pontryagin minimum principle and there is no other relaxed controls with the same value of the objective functional, then this relaxed control is globally optimal.; Finally, we demonstrate that for some generic class of free time optimal control problems for almost all initial points there exists a unique optimal control satisfying the non-degenerated minimum principle. This implies that for such problems our sufficient global optimality conditions can be applied for almost all initial points.