11 research outputs found
A Decomposition Algorithm for Nested Resource Allocation Problems
We propose an exact polynomial algorithm for a resource allocation problem
with convex costs and constraints on partial sums of resource consumptions, in
the presence of either continuous or integer variables. No assumption of strict
convexity or differentiability is needed. The method solves a hierarchy of
resource allocation subproblems, whose solutions are used to convert
constraints on sums of resources into bounds for separate variables at higher
levels. The resulting time complexity for the integer problem is , and the complexity of obtaining an -approximate
solution for the continuous case is , being
the number of variables, the number of ascending constraints (such that ), a desired precision, and the total resource. This
algorithm attains the best-known complexity when , and improves it when
. Extensive experimental analyses are conducted with four
recent algorithms on various continuous problems issued from theory and
practice. The proposed method achieves a higher performance than previous
algorithms, addressing all problems with up to one million variables in less
than one minute on a modern computer.Comment: Working Paper -- MIT, 23 page
Reflection methods for user-friendly submodular optimization
Recently, it has become evident that submodularity naturally captures widely
occurring concepts in machine learning, signal processing and computer vision.
Consequently, there is need for efficient optimization procedures for
submodular functions, especially for minimization problems. While general
submodular minimization is challenging, we propose a new method that exploits
existing decomposability of submodular functions. In contrast to previous
approaches, our method is neither approximate, nor impractical, nor does it
need any cumbersome parameter tuning. Moreover, it is easy to implement and
parallelize. A key component of our method is a formulation of the discrete
submodular minimization problem as a continuous best approximation problem that
is solved through a sequence of reflections, and its solution can be easily
thresholded to obtain an optimal discrete solution. This method solves both the
continuous and discrete formulations of the problem, and therefore has
applications in learning, inference, and reconstruction. In our experiments, we
illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013
Approximation schemes for scheduling on a single machine subject to cumulative deterioration and maintenance
We consider a scheduling problem on a single machine to minimize the makespan. The processing conditions are subject to cumulative deterioration, but can be restored by a single maintenance. We link the problem to the Subset-sum problem (if the duration of maintenance is constant) and to the Half-Product Problem (if the duration of maintenance depends on its start time). For both versions of the problem, we adapt the existing fully polynomial-time approximation schemes to our problems by handling the additive constants
Network Flow Algorithms for Structured Sparsity
We consider a class of learning problems that involve a structured
sparsity-inducing norm defined as the sum of -norms over groups of
variables. Whereas a lot of effort has been put in developing fast optimization
methods when the groups are disjoint or embedded in a specific hierarchical
structure, we address here the case of general overlapping groups. To this end,
we show that the corresponding optimization problem is related to network flow
optimization. More precisely, the proximal problem associated with the norm we
consider is dual to a quadratic min-cost flow problem. We propose an efficient
procedure which computes its solution exactly in polynomial time. Our algorithm
scales up to millions of variables, and opens up a whole new range of
applications for structured sparse models. We present several experiments on
image and video data, demonstrating the applicability and scalability of our
approach for various problems.Comment: accepted for publication in Adv. Neural Information Processing
Systems, 201
A Duality Theory with Zero Duality Gap for Nonlinear Programming
Duality is an important notion for constrained optimization which provides a theoretical foundation for a number of constraint decomposition schemes such as separable programming and for deriving lower bounds in space decomposition algorithms such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex optimization problems, especially discrete and mixed-integer problems where the feasible sets are nonconvex. In this paper, we propose a novel extended duality theory for nonlinear optimization that overcomes some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed-integer spaces under mild conditions
Convex Analysis and Optimization with Submodular Functions: a Tutorial
Set-functions appear in many areas of computer science and applied
mathematics, such as machine learning, computer vision, operations research or
electrical networks. Among these set-functions, submodular functions play an
important role, similar to convex functions on vector spaces. In this tutorial,
the theory of submodular functions is presented, in a self-contained way, with
all results shown from first principles. A good knowledge of convex analysis is
assumed
About strongly polynomial time algorithms for quadratic optimization over submodular constraints
We present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints. The main results are: an O(NM log(N2/M)) algorithm for the problem Network defined on a network on M arcs and N nodes; an O(n log n) algorithm for the tree problem on n variables; an O(n log n) algorithm for the Nested problem,
and a linear time algorithm for the Generalized Upper Bound problem. These algorithms are the best known so rar for these problems. The status of the general problem and open questions are presented as weil.This research has been supported in part by ONR grant N00014-91-J-1241
An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure
We present an exploration of the rich theoretical connections between several
classes of regularized models, network flows, and recent results in submodular
function theory. This work unifies key aspects of these problems under a common
theory, leading to novel methods for working with several important models of
interest in statistics, machine learning and computer vision.
In Part 1, we review the concepts of network flows and submodular function
optimization theory foundational to our results. We then examine the
connections between network flows and the minimum-norm algorithm from
submodular optimization, extending and improving several current results. This
leads to a concise representation of the structure of a large class of pairwise
regularized models important in machine learning, statistics and computer
vision.
In Part 2, we describe the full regularization path of a class of penalized
regression problems with dependent variables that includes the graph-guided
LASSO and total variation constrained models. This description also motivates a
practical algorithm. This allows us to efficiently find the regularization path
of the discretized version of TV penalized models. Ultimately, our new
algorithms scale up to high-dimensional problems with millions of variables
A Class of Convex Quadratic Nonseparable Resource Allocation Problems with Generalized Bound Constraints
We study a convex quadratic nonseparable resource allocation problem that arises in the area of decentralized energy management (DEM), where unbalance in electricity networks has to be minimized. In this problem, the given resource is allocated over a set of activities that is divided into subsets, and a cost is assigned to the overall allocated amount of resources to activities within the same subset. We derive two efficient algorithms with worst-case time complexity to solve this problem. For the special case where all subsets have the same size, one of these algorithms even runs in linear time given the subset size. Both algorithms are inspired by well-studied breakpoint search methods for separable convex resource allocation problems. Numerical evaluations on both real and synthetic data confirm the theoretical efficiency of both algorithms and demonstrate their suitability for integration in DEM systems