Abstract:The work of this paper involves two aspects of mathematics and mechanics. In terms of mathematics: (1) The reciprocal programming newly proposed in the mathematical programming theory is developed from the s-m type (or called m-s type) to the s-s type and m-m type reciprocal programming (among which s means single goal, m means multiple goals), so that the definition of reciprocal programming is complete into three types; (2) From the KKT condition to examine the two aspects of reciprocal programming, it is obtained that the three theorems of two sides of reciprocal programming which involves quasi-isomorphism theorem and quasi-simultaneous solution theorems. Moreover, the proofs of the three theorems provide a theoretical basis for the two parties to reference and compare from each other in the solution of reciprocal programming; (3) Respectively, the solution strategies and detailed solutions are given for the case where both sides of a pair of reciprocal programming are reasonable ,and the case where one aspect is unreasonable while the other is reasonable. In terms of mechanics: (1) This paper gives the explanation of whether the structural optimization design model is reasonable or not; (2) In the application of reciprocal programming to structural topology optimization, a solution strategy for unreasonable structural topology optimization models is proposed; (3) A way to solve the MCVC model (the minimization of multiple compliances under a given volume) with the help of the MVCC model (Minimization of the volume under multiple compliance constraints), where the modeling is based on the ICM (Independent Continuous Mapping) method. At the end of this article, four numerical examples are given by Matlab codes, where two of the mathematical programming problems illustrate the relationship between the two sides of reciprocal programming, and two of structural topology optimization problems is two cases in many structural topology optimization problems. The numerical results support the reciprocal programming theory proposed in this paper.