A recurrence method for a special class of continuous time linear programming problems
Journal of Global Optimization
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In this thesis we study the theory and algorithms for separated continuous linear programming (SCLP) and its extensions. We first investigate the relationships among SCLP, the dual of SCLP and the corresponding discretized versions of them. By using the symmetric primal and dual structure and an even partition of the time interval [0, T], we show that the strong duality holds between SCLP and its dual problem under some mild assumption. This is actually an alternative proof for the strong duality theorem. The other constructive proof is due to Weiss [50]. Our new proof is more direct and can be easily extended to prove the same strong duality results for the extensions of SCLP. Based on these results, we propose an approximation algorithm which solves SCLP with any prescribed precision requirement. Our algorithm is in fact a polynomial-time approximation (PTA) scheme. The trade-off between the quality of the solution and the computational effort is explicit. We then study the extensions of SCLP; that is, separated continuous conic programming (SCCP) and its generalized version (GSCCP). It turns out that our results on SCLP can be readily extended to SCCP and GSCCP. To our knowledge, SCCP and GSCCP are new models with novel applications. Throughout this thesis, some numerical examples are used to illustrate the algorithms that we propose. In particular, we solve a special LQ control problem with sign constraints on the state and the control variables as an instance of SCCP, yielding a new solution method for such kind of LQ control problems.