Effective method for constrained minimum — reverse bridge theorem

We propose reverse bridge theorem (RBTH) and give the demonstration in this work. RBTH is an effective method for constrained minimum. It is necessary and sufficient condition. RBTH bases on Extended saddle-point condition (ESPC), which is proposed by Yixin Chen. SGPlan according to ESPC dose the best in satisfying the International Planning Competition 2006 and in Suboptimal Metric Track of International Planning Competition 2004. RBTH comes into all advantage of ESPC and is different from ESPC, which can be concluded from two points. First, the core inequality of ESPC is formed by two sub-inequalities, and the core inequality of RBTH contains only one sub-inequality and one sub-equality. Generally, equality is easier to handle than inequality. RBTH should be better for planner than ESPC. The core inequality changes and is no longer saddle shape. We analyze why the core inequality changes. Second, continuous RBTH (CRBTH) does not need constraint-qualification condition which is necessary for continuous ESPC. This can be authenticated in two respects. One is the proof of CRBTH; the other is KKT which has relationship with ESPC. RBTH is real necessary and sufficient condition and ESPC is not. So RBTH can be used more widely. Besides, we show some errors in ESPC. We define point domain and Shrink-point domain to explain error. We also analyze the cause of errors, and provide an example.