MPC

MPC 2019, ISSUE 4



Mathematical Programming Computation, Volume 11, Issue 4, December 2019

A derivative-free Gauss–Newton method

Coralia Cartis, Lindon Roberts

We present DFO-GN, a derivative-free version of the Gauss–Newton method for solving nonlinear least-squares problems. DFO-GN uses linear interpolation of residual values to build a quadratic model of the objective, which is then used within a typical derivative-free trust-region framework. We show that DFO-GN is globally convergent and requires at most O(??2) iterations to reach approximate first-order criticality within tolerance ?. We provide an implementation of DFO-GN and compare it to other state-of-the-art derivative-free solvers that use quadratic interpolation models. We demonstrate numerically that despite using only linear residual models, DFO-GN performs comparably to these methods in terms of objective evaluations. Furthermore, as a result of the simplified interpolation procedure, DFO-GN has superior runtime and scalability. Our implementation of DFO-GN is available at https://github.com/numericalalgorithmsgroup/dfogn (https://doi.org/10.5281/zenodo.2629875).

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Mathematical Programming Computation, Volume 11, Issue 4, December 2019

Structure-driven fix-and-propagate heuristics for mixed integer programming

Gerald Gamrath, Timo Berthold, Stefan Heinz, Michael Winkler

Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They often provide good feasible solutions early and help to reduce the time needed to prove optimality. In this paper, we present a scheme for start heuristics that can be executed without previous knowledge of an LP solution or a previously found integer feasible solution. It uses global structures available within MIP solvers to iteratively fix integer variables and propagate these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. If sufficiently many variables can be fixed that way, the resulting problem is solved first as an LP, and then as an auxiliary MIP if the rounded LP solution does not provide a feasible solution already. We present three primal heuristics that use this scheme based on different global structures. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about 60% of the instances and by this, help to improve several performance measures for MIP solvers, including the primal integral and the average solving time.

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Mathematical Programming Computation, Volume 11, Issue 4, December 2019

New exact approaches to row layout problems

Anja Fischer Fischer, Frank Fischer, Philipp Hungerländer

Given a set of departments, a number of rows and pairwise connectivities between these departments, the multi-row facility layout problem (MRFLP) looks for a non-overlapping arrangement of these departments in the rows such that the weighted sum of the center-to-center distances is minimized. As even small instances of the MRFLP are rather challenging, several special cases have been considered in the literature. In this paper we present new mixed-integer linear programming formulations for the (space-free) multi-row facility layout problem with given assignment of the departments to the rows that combine distance and betweenness variables. Using these formulations instances with up to 25 departments can be solved to optimality (within at most 6 h) for the first time. Furthermore, we are able to reduce the running times for instances with up to 23 departments significantly in comparison to the literature. Later on we use these formulations in an enumeration scheme for solving the (space-free) multi-row facility layout problem. In particular, we test all possible row assignments, where some assignments are excluded due to our new combinatorial investigations. For the first time this approach enables us to solve instances with two rows with up to 16 departments, with three rows with up to 15 departments and with four and five rows with up to 13 departments exactly in reasonable time.

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Mathematical Programming Computation, Volume 11, Issue 4, December 2019

An active set algorithm for robust combinatorial optimization based on separation oracles

Christoph Buchheim, Marianna De Santis

We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurobi.

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