873 research outputs found
Learning to Pivot as a Smart Expert
Linear programming has been practically solved mainly by simplex and interior
point methods. Compared with the weakly polynomial complexity obtained by the
interior point methods, the existence of strongly polynomial bounds for the
length of the pivot path generated by the simplex methods remains a mystery. In
this paper, we propose two novel pivot experts that leverage both global and
local information of the linear programming instances for the primal simplex
method and show their excellent performance numerically. The experts can be
regarded as a benchmark to evaluate the performance of classical pivot rules,
although they are hard to directly implement. To tackle this challenge, we
employ a graph convolutional neural network model, trained via imitation
learning, to mimic the behavior of the pivot expert. Our pivot rule, learned
empirically, displays a significant advantage over conventional methods in
various linear programming problems, as demonstrated through a series of
rigorous experiments
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Conic Sampling: An Efficient Method for Solving Linear and Quadratic Programming by Randomly Linking Constraints within the Interior
Linear programming (LP) problems are commonly used in analysis and resource allocation, frequently surfacing as approximations to more difficult problems. Existing approaches to LP have been dominated by a small group of methods, and randomized algorithms have not enjoyed popularity in practice. This paper introduces a novel randomized method of solving LP problems by moving along the facets and within the interior of the polytope along rays randomly sampled from the polyhedral cones defined by the bounding constraints. This conic sampling method is then applied to randomly sampled LPs, and its runtime performance is shown to compare favorably to the simplex and primal affine-scaling algorithms, especially on polytopes with certain characteristics. The conic sampling method is then adapted and applied to solve a certain quadratic program, which compute a projection onto a polytope; the proposed method is shown to outperform the proprietary software Mathematica on large, sparse QP problems constructed from mass spectometry-based proteomics
Experimental Geometry Optimization Techniques for Multi-Element Airfoils
A study is reported on geometry optimization techniques for high-lift airfoils. A modern three-element airfoil model with a remotely actuated flap was designed, tested, and used in wind tunnel experiments to investigate optimum flap positioning based on lift. All the results presented were obtained in the Old Dominion University low-speed wind tunnel. Detailed results for lift coefficient versus flap vertical and horizontal position are presented for two airfoil angles-of-attack: 8 and 14 degrees. Three automated optimization simulations, the method of steepest ascent and two variants of the sequential simplex method, were demonstrated using experimental data. An on-line optimizer was demonstrated with the wind tunnel model which automatically seeks the optimum lift as a function of flap position. Hysteresis in lift as a function of flap position was discovered when tests were conducted with continuous flow conditions. It was shown that optimum lift coefficients determined using continuous flow conditions exist over an extended range of flap positions when compared to those determined using traditional intermittent conditions
A unified worst case for classical simplex and policy iteration pivot rules
We construct a family of Markov decision processes for which the policy
iteration algorithm needs an exponential number of improving switches with
Dantzig's rule, with Bland's rule, and with the Largest Increase pivot rule.
This immediately translates to a family of linear programs for which the
simplex algorithm needs an exponential number of pivot steps with the same
three pivot rules. Our results yield a unified construction that simultaneously
reproduces well-known lower bounds for these classical pivot rules, and we are
able to infer that any (deterministic or randomized) combination of them cannot
avoid an exponential worst-case behavior. Regarding the policy iteration
algorithm, pivot rules typically switch multiple edges simultaneously and our
lower bound for Dantzig's rule and the Largest Increase rule, which perform
only single switches, seem novel. Regarding the simplex algorithm, the
individual lower bounds were previously obtained separately via deformed
hypercube constructions. In contrast to previous bounds for the simplex
algorithm via Markov decision processes, our rigorous analysis is reasonably
concise
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