Computational geometry: an introduction
Computational geometry: an introduction
Convexity and concavity properties of the optimal value function in parametric nonlinear programming
Journal of Optimization Theory and Applications
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Approximating the Pareto-hull of a convex set by polyhedral sets
Computational Mathematics and Mathematical Physics
Convex Optimization
Multicriteria Optimization
An approximation algorithm for convex multi-objective programming problems
Journal of Global Optimization
Enhancement of Sandwich Algorithms for Approximating Higher-Dimensional Convex Pareto Sets
INFORMS Journal on Computing
A dual variant of Benson's "outer approximation algorithm" for multiple objective linear programming
Journal of Global Optimization
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We consider the problem of approximating Pareto surfaces of convex multicriteria optimization problems by a discrete set of points and their convex combinations. Finding the scalarization parameters that optimally limit the approximation error when generating a single Pareto optimal solution is a nonconvex optimization problem. This problem can be solved by enumerative techniques but at a cost that increases exponentially with the number of objectives. We present an algorithm for solving the Pareto surface approximation problem that is practical with 10 or less conflicting objectives, motivated by an application to radiation therapy optimization. Our enumerative scheme is, in a sense, dual to a family of previous algorithms. The proposed technique retains the quality of the best previous algorithm in this class while solving fewer subproblems. A further improvement is provided by a procedure for discarding subproblems based on reusing information from previous solves. The combined effect of the enhancements is empirically demonstrated to reduce the computational expense of solving the Pareto surface approximation problem by orders of magnitude. For problems where the objectives have positive curvature, an improved bound on the approximation error is demonstrated using transformations of the initial objectives with strictly increasing and concave functions.