We propose an SQP type method for constrained nonlinear multiobjective optimization. The proposed algorithm maintains a list of nondominated points that is improved both for spread along the Pareto front and optimality by solving singleobjective constrained optimization problems. Under appropriate differentiability assumptions we discuss convergence to local optimal Pareto points. We provide numerical results for … Read more
This paper deals with the problem of finding convex bulges on the Pareto-front of a multi-objective optimization problem. The point of maximum bulge is of particular interest as this point shows good trade-off properties and it is also close to the non-attainable utopia point. Our approach is to use a population based algorithm to simultaneously … Read more
The types of constraints encountered in black-box simulation-based optimization problems differ significantly from those addressed in nonlinear programming. We introduce a characterization of constraints to address this situation. We provide formal definitions for several constraint classes and present illustrative examples in the context of the resulting taxonomy. This taxonomy, denoted KARQ, is useful for modeling … Read more
Variable selection is a fundamental task in statistical data analysis. Sparsity-inducing regularization methods are a popular class of methods that simultaneously perform variable selection and model estimation. The central problem is a quadratic optimization problem with an $\ell_0$-norm penalty. Exactly enforcing the $\ell_0$-norm penalty is computationally intractable for larger scale problems, so different sparsity-inducing penalty … Read more
For problems when decisions are taken prior to observing the realization of underlying random events, probabilistic constraints are an important modelling tool if reliability is a concern. A key concept to numerically dealing with probabilistic constraints is that of p-efficient points. By adopting a dual point of view, we develop a solution framework that includes … Read more
The alternating direction method of multipliers (ADMM) is an application of the Douglas-Rachford splitting method; and the symmetric version of ADMM which updates the Lagrange multiplier twice at each iteration is an application of the Peaceman-Rachford splitting method. Sometimes the symmetric ADMM works empirically; but theoretically its convergence is not guaranteed. It was recently found … Read more
In this paper, firstly, the concept of $\varepsilon-$strictly efficient subdifferential for set-valued map is introduced in Hausdorff locally convex topological vector spaces. Secondly, a characterization of this subdifferential by scalarization and the generalized $\varepsilon-$ Moreau-Rockafellar type theorem for set-valued maps are established. Finally, the necessary optimality condition of the constraint set-valued optimization problem for $\varepsilon-$ … Read more
The “sequential ordering problem” (SOP) is the generalisation of the asymmetric travelling salesman problem in which there are precedence relations between pairs of nodes. Hernández & Salazar introduced a “multi-commodity flow” (MCF) formulation for a generalisation of the SOP in which the vehicle has a limited capacity. We strengthen this MCF formulation by fixing variables … Read more
We develop a new constraint-reduced infeasible predictor-corrector interior point method for semidefinite programming, and we prove that it has polynomial global convergence and superlinear local convergence. While the new algorithm uses HKM direction in predictor step, it adopts AHO direction in corrector step to obtain faster approach to the central path. In contrast to the … Read more
We consider the linear or quadratic 0/1 program \[P:\quad f^*=\min\{ c^Tx+x^TFx : \:A\,x =\b;\:x\in\{0,1\}^n\},\] for some vectors $c\in R^n$, $b\in Z^m$, some matrix $A\in Z^{m\times n}$ and some real symmetric matrix $F\in R^{n\times n}$. We show that $P$ can be formulated as a MAX-CUT problem whose quadratic form criterion is explicit from the data of … Read more