15,558 research outputs found
Disciplined Quasiconvex Programming
We present a composition rule involving quasiconvex functions that
generalizes the classical composition rule for convex functions. This rule
complements well-known rules for the curvature of quasiconvex functions under
increasing functions and pointwise maximums. We refer to the class of
optimization problems generated by these rules, along with a base set of
quasiconvex and quasiconcave functions, as disciplined quasiconvex programs.
Disciplined quasiconvex programming generalizes disciplined convex programming,
the class of optimization problems targeted by most modern domain-specific
languages for convex optimization. We describe an implementation of disciplined
quasiconvex programming that makes it possible to specify and solve quasiconvex
programs in CVXPY 1.0.Comment: p. 4: corrected typo
Evolutionary Optimization for Decision Making under Uncertainty
Optimizing decision problems under uncertainty can be done using a variety of
solution methods. Soft computing and heuristic approaches tend to be powerful
for solving such problems. In this overview article, we survey Evolutionary
Optimization techniques to solve Stochastic Programming problems - both for the
single-stage and multi-stage case.Comment: Keynote talk at the MENDEL 201
Column generation based math-heuristic for classification trees
This paper explores the use of Column Generation (CG) techniques in
constructing univariate binary decision trees for classification tasks. We
propose a novel Integer Linear Programming (ILP) formulation, based on
root-to-leaf paths in decision trees. The model is solved via a Column
Generation based heuristic. To speed up the heuristic, we use a restricted
instance data by considering a subset of decision splits, sampled from the
solutions of the well-known CART algorithm. Extensive numerical experiments
show that our approach is competitive with the state-of-the-art ILP-based
algorithms. In particular, the proposed approach is capable of handling big
data sets with tens of thousands of data rows. Moreover, for large data sets,
it finds solutions competitive to CART
Convergence of Weighted Min-Sum Decoding Via Dynamic Programming on Trees
Applying the max-product (and belief-propagation) algorithms to loopy graphs
is now quite popular for best assignment problems. This is largely due to their
low computational complexity and impressive performance in practice. Still,
there is no general understanding of the conditions required for convergence
and/or the optimality of converged solutions. This paper presents an analysis
of both attenuated max-product (AMP) decoding and weighted min-sum (WMS)
decoding for LDPC codes which guarantees convergence to a fixed point when a
weight parameter, {\beta}, is sufficiently small. It also shows that, if the
fixed point satisfies some consistency conditions, then it must be both the
linear-programming (LP) and maximum-likelihood (ML) solution.
For (dv,dc)-regular LDPC codes, the weight must satisfy {\beta}(dv-1) \leq 1
whereas the results proposed by Frey and Koetter require instead that
{\beta}(dv-1)(dc-1) < 1. A counterexample which shows a fixed point might not
be the ML solution if {\beta}(dv-1) > 1 is also given. Finally, connections are
explored with recent work by Arora et al. on the threshold of LP decoding.Comment: 43 pages, 3 figure
On Decoding Irregular Tanner Codes with Local-Optimality Guarantees
We consider decoding of binary Tanner codes using message-passing iterative
decoding and linear programming (LP) decoding in MBIOS channels. We present new
certificates that are based on a combinatorial characterization for
local-optimality of a codeword in irregular Tanner codes with respect to any
MBIOS channel. This characterization is based on a conical combination of
normalized weighted subtrees in the computation trees of the Tanner graph.
These subtrees may have any finite height h (even equal or greater than half of
the girth of the Tanner graph). In addition, the degrees of local-code nodes in
these subtrees are not restricted to two. We prove that local optimality in
this new characterization implies maximum-likelihood (ML) optimality and LP
optimality, and show that a certificate can be computed efficiently.
We also present a new message-passing iterative decoding algorithm, called
normalized weighted min-sum (NWMS). NWMS decoding is a BP-type algorithm that
applies to any irregular binary Tanner code with single parity-check local
codes. We prove that if a locally-optimal codeword with respect to height
parameter h exists (whereby notably h is not limited by the girth of the Tanner
graph), then NWMS decoding finds this codeword in h iterations. The decoding
guarantee of the NWMS decoding algorithm applies whenever there exists a
locally optimal codeword. Because local optimality of a codeword implies that
it is the unique ML codeword, the decoding guarantee also provides an ML
certificate for this codeword.
Finally, we apply the new local optimality characterization to regular Tanner
codes, and prove lower bounds on the noise thresholds of LP decoding in MBIOS
channels. When the noise is below these lower bounds, the probability that LP
decoding fails decays doubly exponentially in the girth of the Tanner graph
Efficient evaluation of mp-MIQP solutions using lifting
This paper presents an efficient approach for the evaluation of
multi-parametric mixed integer quadratic programming (mp-MIQP) solutions,
occurring for instance in control problems involving discrete time hybrid
systems with quadratic cost. Traditionally, the online evaluation requires a
sequential comparison of piecewise quadratic value functions. As the main
contribution, we introduce a lifted parameter space in which the piecewise
quadratic value functions become piecewise affine and can be merged to a single
value function defined over a single polyhedral partition without any overlaps.
This enables efficient point location approaches using a single binary search
tree. Numerical experiments include a power electronics application and
demonstrate an online speedup up to an order of magnitude. We also show how the
achievable online evaluation time can be traded off against the offline
computational time.Comment: 23 pages, update includes more details Theorem 1 proo
Finding Minimum Spanning Forests in a Graph
We introduce a graph partitioning problem motivated by computational topology
and propose two algorithms that produce approximate solutions. Specifically,
given a weighted, undirected graph and a positive integer , we desire to
find disjoint trees within such that each vertex of is contained in
one of the trees and the weight of the largest tree is as small as possible. We
are unable to find this problem in the graph partitioning literature, but we
show that the problem is NP-complete. We then propose two approximation
algorithms, one that uses a spectral clustering approach and another that
employs a dynamic programming strategy, which produce near-optimal partitions
on a family of test graphs. We describe these algorithms and analyze their
empirical performance.Comment: 13 page
Inference in Graphical Models via Semidefinite Programming Hierarchies
Maximum A posteriori Probability (MAP) inference in graphical models amounts
to solving a graph-structured combinatorial optimization problem. Popular
inference algorithms such as belief propagation (BP) and generalized belief
propagation (GBP) are intimately related to linear programming (LP) relaxation
within the Sherali-Adams hierarchy. Despite the popularity of these algorithms,
it is well understood that the Sum-of-Squares (SOS) hierarchy based on
semidefinite programming (SDP) can provide superior guarantees. Unfortunately,
SOS relaxations for a graph with vertices require solving an SDP with
variables where is the degree in the hierarchy. In
practice, for , this approach does not scale beyond a few tens of
variables. In this paper, we propose binary SDP relaxations for MAP inference
using the SOS hierarchy with two innovations focused on computational
efficiency. Firstly, in analogy to BP and its variants, we only introduce
decision variables corresponding to contiguous regions in the graphical model.
Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style
method, and develop a sequential rounding procedure. We demonstrate that the
resulting algorithm can solve problems with tens of thousands of variables
within minutes, and outperforms BP and GBP on practical problems such as image
denoising and Ising spin glasses. Finally, for specific graph types, we
establish a sufficient condition for the tightness of the proposed partial SOS
relaxation
Constrained clustering via diagrams: A unified theory and its applications to electoral district design
The paper develops a general framework for constrained clustering which is
based on the close connection of geometric clustering and diagrams. Various new
structural and algorithmic results are proved (and known results generalized
and unified) which show that the approach is computationally efficient and
flexible enough to pursue various conflicting demands.
The strength of the model is also demonstrated practically on real-world
instances of the electoral district design problem where municipalities of a
state have to be grouped into districts of nearly equal population while
obeying certain politically motivated requirements
Twenty (or so) Questions: -ary Length-Bounded Prefix Coding
Efficient optimal prefix coding has long been accomplished via the Huffman
algorithm. However, there is still room for improvement and exploration
regarding variants of the Huffman problem. Length-limited Huffman coding,
useful for many practical applications, is one such variant, for which codes
are restricted to the set of codes in which none of the codewords is longer
than a given length, . Binary length-limited coding can be done in
time and O(n) space via the widely used Package-Merge algorithm
and with even smaller asymptotic complexity using a lesser-known algorithm. In
this paper these algorithms are generalized without increasing complexity in
order to introduce a minimum codeword length constraint , to allow
for objective functions other than the minimization of expected codeword
length, and to be applicable to both binary and nonbinary codes; nonbinary
codes were previously addressed using a slower dynamic programming approach.
These extensions have various applications -- including fast decompression and
a modified version of the game ``Twenty Questions'' -- and can be used to solve
the problem of finding an optimal code with limited fringe, that is, finding
the best code among codes with a maximum difference between the longest and
shortest codewords. The previously proposed method for solving this problem was
nonpolynomial time, whereas solving this using the novel linear-space algorithm
requires only time, or even less if is not .Comment: 12 pages, 4 figures, extended version of cs/0701012 (accepted to ISIT
2007), formerly "Twenty (or so) Questions: -ary Bounded-Length Huffman
Coding
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