The multiobjective bilevel program is a sequence of two optimization problems where the upper level problem is multiobjective and the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. In the case where the Karush-Kuhn-Tucker (KKT) condition is necessary and sufficient for global optimality of … Read more
The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behaviour of the objective function. Following earlier work by Gratton and Toint (2009), we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take … Read more
We consider Hessian approximation schemes for large-scale multilevel unconstrained optimization problems, which typically present a sparsity and partial separability structure. This allows iterative quasi-Newton methods to solve them despite of their size. Structured finite-difference methods and updating schemes based on the secant equation are presented and compared numerically inside the multilevel trust-region algorithm proposed by … Read more
We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton, Mouffe, Toint, Weber (2008) for bound-constrained nonlinear problems, and provide numerical experience on multilevel test problems. A suitable choice of the algorithm’s parameters is identified on these problems, yielding a satisfactory compromise between reliability and efficiency. The resulting default algorithm is then … Read more
Partly due to lack of test problems, the impact of the Pareto set (PS) shapes on the performance of evolutionary algorithms has not yet attracted much attention. This paper introduces a general class of continuous multiobjective optimization test instances with arbitrary prescribed PS shapes, which could be used for studying the ability of MOEAs for … Read more
A recursive trust-region method is introduced for the solution of bound-constrained nonlinear nonconvex optimization problems for which a hierarchy of descriptions exists. Typical cases are infinite-dimensional problems for which the levels of the hierarchy correspond to discretization levels, from coarse to fine. The new method uses the infinity norm to define the shape of the … Read more
We present a new primal-dual algorithm for convex quadratic multicriteria optimization. The algorithm is able to adaptively refine the approximation to the set of efficient points by way of a warm-start interior-point scalarization approach. Results of this algorithm when applied on a three-criteria real-world power plant optimization problem are reported, thereby illustrating the feasibility of … Read more
We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton, Sartenaer, Toint (2004), and provide significant numerical experience on multilevel test problems. A suitable choice of the algorithm’s parameters is identified on these problems, yielding a very satisfactory compromise between reliability and efficiency. The resulting default algorithm is then compared to alternative … Read more
Convergence properties of trust-region methods for unconstrained nonconvex optimization is considered in the case where information on the objective function’s local curvature is incomplete, in the sense that it may be restricted to a fixed set of “test directions” and may not be available at every iteration. It is shown that convergence to local “weak” … Read more
We propose a new approach to convex nonlinear multiobjective optimization that captures the geometry of the Pareto set by generating a discrete set of Pareto points optimally. We show that the problem of finding an optimal representation of the Pareto surface can be formulated as a mathematical program with complementarity constraints. The complementarity constraints arise … Read more