A sequential convex programming algorithm for minimizing a sum of Euclidean norms with non-convex constraints

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 varia...

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Bibliographic Details
Published in:Optimization methods & software Vol. 31; no. 1; pp. 187 - 203
Main Authors: Trang, L.H., Kozma, A., An, P.T., Diehl, M.
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 02.01.2016
Taylor & Francis Ltd
Subjects:
Algorithms
Approximation
Boundaries
Mathematical analysis
Mathematical models
Mathematical programming
Matlab
minimizing a sum of Euclidean norms
non-convex constraint
Optimization
relaxation
sequentialconvex programming
shortest path
ISSN:1055-6788, 1029-4937
Online Access:Get full text
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Summary: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.
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ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2015.1055561