A sequential convex programming algorithm for minimizing a sum of Euclidean norms with non-convex constraints - Trang, L.H.; Kozma, A.; An, P.T.; Diehl, M. (Optimization methods & software, 2016)
Given and a finite set of convex polygons in , we consider the problem of finding the Euclidean shortest path starting at p then visiting the relative boundaries of the convex polygons in a given order, and ending at q. An approximate algorithm is proposed. The problem can be rewritten under a variant of minimizing a sum of Euclidean norms: , where and , subject to is on the relative boundary of , for . The objective function of the problem is convex but not everywhere differentiable and the constraints are non-convex. By using a smooth inner approximation of with parameter t, a relaxed form of the problem is constructed such that its solution, denoted by , is inside but outside the inner approximation. The relaxed problem is then solved iteratively using a sequential convex programming. The obtained solution , however, is actually not on the relative boundary of . Then a so-called refinement of is finally required to determine a solution passing through the relative boundary of , for . It is shown that the solution of the relaxed problem tends to its refined one as . The algorithm is implemented in Matlab using the CVX package. Numerical tests indicate that the solution obtained by the algorithm is close to the global one.