87,219 research outputs found
On the dimension of max-min convex sets
We introduce a notion of dimension of max-min convex sets, following the
approach of tropical convexity. We introduce a max-min analogue of the tropical
rank of a matrix and show that it is equal to the dimension of the associated
polytope. We describe the relation between this rank and the notion of strong
regularity in max-min algebra, which is traditionally defined in terms of
unique solvability of linear systems and trapezoidal property.Comment: 19 pages, v2: many corrections in the proof
Min-max-min robust combinatorial optimization
In this thesis we introduce a robust optimization approach which is based on a binary min-max-min problem. The so called Min-max-min Robust Optimization extends the classical min-max approach by calculating k different solutions instead of one.
Usually in robust optimization we consider problems whose problem parameters can be uncertain. The basic idea is to define an uncertainty set U which contains all relevant problem parameters, called scenarios. The objective is then to calculate a solution which is feasible for every scenario in U and which optimizes the worst objective value over all scenarios in U.
As a special case of the K-adaptability approach for robust two-stage problems, the min-max-min robust optimization approach aims to calculate k different solutions for the underlying combinatorial problem, such that, considering the best of these solutions in each scenario, the worst objective value over all scenarios is optimized. This idea can be modeled as a min-max-min problem.
In this thesis we analyze the complexity of the afore mentioned problem for convex and for discrete uncertainty sets U. We will show that under further assumptions the problem is as easy as the underlying combinatorial problem for convex uncertainty sets if the number of calculated solutions is greater than the dimension of the problem. Additionally we present a practical exact algorithm to solve the min-max-min problem for any combinatorial problem, given by a deterministic oracle. On the other hand we prove that if we fix the number of solutions k, then the problem is NP-hard even for polyhedral uncertainty sets and the unconstrained binary problem. For the case when the number of calculated solutions is lower or equal to the dimension we present a heuristic algorithm which is based on the exact algorithm above. Both algorithms are tested and analyzed on random instances of the knapsack problem, the vehicle routing problem and the shortest path problem.
For discrete uncertainty sets we show that the min-max-min problem is NP-hard for a selection of combinatorial problems. Nevertheless we prove that it can be solved in pseudopolynomial time or admits an FPTAS if the min-max problem can be solved in pseudopolynomial or admits an FPTAS respectively
The convexification effect of Minkowski summation
Let us define for a compact set the sequence It was independently proved by Shapley, Folkman and Starr (1969)
and by Emerson and Greenleaf (1969) that approaches the convex hull of
in the Hausdorff distance induced by the Euclidean norm as goes to
. We explore in this survey how exactly approaches the convex
hull of , and more generally, how a Minkowski sum of possibly different
compact sets approaches convexity, as measured by various indices of
non-convexity. The non-convexity indices considered include the Hausdorff
distance induced by any norm on , the volume deficit (the
difference of volumes), a non-convexity index introduced by Schneider (1975),
and the effective standard deviation or inner radius. After first clarifying
the interrelationships between these various indices of non-convexity, which
were previously either unknown or scattered in the literature, we show that the
volume deficit of does not monotonically decrease to 0 in dimension 12
or above, thus falsifying a conjecture of Bobkov et al. (2011), even though
their conjecture is proved to be true in dimension 1 and for certain sets
with special structure. On the other hand, Schneider's index possesses a strong
monotonicity property along the sequence , and both the Hausdorff
distance and effective standard deviation are eventually monotone (once
exceeds ). Along the way, we obtain new inequalities for the volume of the
Minkowski sum of compact sets, falsify a conjecture of Dyn and Farkhi (2004),
demonstrate applications of our results to combinatorial discrepancy theory,
and suggest some questions worthy of further investigation.Comment: 60 pages, 7 figures. v2: Title changed. v3: Added Section 7.2
resolving Dyn-Farkhi conjectur
Effective Condition Number Bounds for Convex Regularization
We derive bounds relating Renegar's condition number to quantities that
govern the statistical performance of convex regularization in settings that
include the -analysis setting. Using results from conic integral
geometry, we show that the bounds can be made to depend only on a random
projection, or restriction, of the analysis operator to a lower dimensional
space, and can still be effective if these operators are ill-conditioned. As an
application, we get new bounds for the undersampling phase transition of
composite convex regularizers. Key tools in the analysis are Slepian's
inequality and the kinematic formula from integral geometry.Comment: 17 pages, 4 figures . arXiv admin note: text overlap with
arXiv:1408.301
On hyperplanes and semispaces in max-min convex geometry
The concept of separation by hyperplanes is fundamental for convex geometry
and its tropical (max-plus) analogue. However, analogous separation results in
max-min convex geometry are based on semispaces. This paper answers the
question which semispaces are hyperplanes and when it is possible to
classically separate by hyperplanes in max-min convex geometry
Reachability analysis of linear hybrid systems via block decomposition
Reachability analysis aims at identifying states reachable by a system within
a given time horizon. This task is known to be computationally expensive for
linear hybrid systems. Reachability analysis works by iteratively applying
continuous and discrete post operators to compute states reachable according to
continuous and discrete dynamics, respectively. In this paper, we enhance both
of these operators and make sure that most of the involved computations are
performed in low-dimensional state space. In particular, we improve the
continuous-post operator by performing computations in high-dimensional state
space only for time intervals relevant for the subsequent application of the
discrete-post operator. Furthermore, the new discrete-post operator performs
low-dimensional computations by leveraging the structure of the guard and
assignment of a considered transition. We illustrate the potential of our
approach on a number of challenging benchmarks.Comment: Accepted at EMSOFT 202
A partial differential equation for the strictly quasiconvex envelope
In a series of papers Barron, Goebel, and Jensen studied Partial Differential
Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions,
barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome
the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE
for \e-robust QC functions, which is well-posed. Building on this work, we
introduce a stronger regularization which is amenable to numerical
approximation. We build convergent finite difference approximations, comparing
the QC envelope and the two regularization. Solutions of this PDE are strictly
convex, and smoother than the robust-QC functions.Comment: 20 pages, 6 figures, 1 tabl
Gordon's inequality and condition numbers in conic optimization
The probabilistic analysis of condition numbers has traditionally been
approached from different angles; one is based on Smale's program in complexity
theory and features integral geometry, while the other is motivated by
geometric functional analysis and makes use of the theory of Gaussian
processes. In this note we explore connections between the two approaches in
the context of the biconic homogeneous feasiblity problem and the condition
numbers motivated by conic optimization theory. Key tools in the analysis are
Slepian's and Gordon's comparision inequalities for Gaussian processes,
interpreted as monotonicity properties of moment functionals, and their
interplay with ideas from conic integral geometry
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