174 research outputs found
An interior-point method for the single-facility location problem with mixed norms using a conic formulation
We consider the single-facility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in R, where each distance can be measured according to a different p-norm.We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving three-dimensional cones. Using the availability of a self-concordant barrier for these cones, we present a polynomial-time algorithm (a long-step path-following interior-point scheme) to solve the problem up to a given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem.nonsymmetric conic optimization, conic reformulation, convex optimization, sum of norm minimization, single-facility location problems, interior-point methods
Projectively self-concordant barriers on convex sets
Self-concordance is the most important property required for barriers in
convex programming. We introduce an alternative, stronger notion, which we call
projective self-concordance, define the corresponding Dikin sets by a quadratic
inequality, and develop a corresponding duality theory. Our notion is
equivariant with respect to the group of projective transformations, which is
larger than the affine group corresponding to the classical notion. Our Dikin
sets are larger than the classical Dikin ellipsoids, depend on the gradient of
the barrier at the center point, are non-symmetric, and may even be unbounded.
From the derivatives of the barrier at a given point we construct a quadratic
set which overbounds the underlying convex set, which is not possible for the
classical notion of self-concordance. This opens the way to design algorithms
which make larger steps and hence have a faster convergence rate than
traditional interior-point methods. We give many examples of convex sets with
projectively self-concordant barriers
Volumetric Spanners: an Efficient Exploration Basis for Learning
Numerous machine learning problems require an exploration basis - a mechanism
to explore the action space. We define a novel geometric notion of exploration
basis with low variance, called volumetric spanners, and give efficient
algorithms to construct such a basis.
We show how efficient volumetric spanners give rise to the first efficient
and optimal regret algorithm for bandit linear optimization over general convex
sets. Previously such results were known only for specific convex sets, or
under special conditions such as the existence of an efficient self-concordant
barrier for the underlying set
Interior Point Methods with a Gradient Oracle
We provide an interior point method based on quasi-Newton iterations, which
only requires first-order access to a strongly self-concordant barrier
function. To achieve this, we extend the techniques of Dunagan-Harvey [STOC
'07] to maintain a preconditioner, while using only first-order information. We
measure the quality of this preconditioner in terms of its relative
excentricity to the unknown Hessian matrix, and we generalize these techniques
to convex functions with a slowly-changing Hessian. We combine this with an
interior point method to show that, given first-order access to an appropriate
barrier function for a convex set , we can solve well-conditioned linear
optimization problems over to precision in time
,
where is the self-concordance parameter of the barrier function, and
is the time required to make a gradient query. As a consequence
we show that:
Linear optimization over -dimensional convex sets can be solved
in time
.
This parallels the running time achieved by state of the art algorithms for
cutting plane methods, when replacing separation oracles with first-order
oracles for an appropriate barrier function.
We can solve semidefinite programs involving matrices in
in time
,
improving over the state of the art algorithms, in the case where
.
Along the way we develop a host of tools allowing us to control the evolution
of our potential functions, using techniques from matrix analysis and Schur
convexity.Comment: STOC 202
Interior point methods and simulated annealing for nonsymmetric conic optimization
This thesis explores four methods for convex optimization. The first two are an interior point method and a simulated annealing algorithm that share a theoretical foundation. This connection is due to the interior point method’s use of the so-called entropic barrier, whose derivatives can be approximated through sampling. Here, the sampling will be carried out with a technique known as hit-and-run. By carefully analyzing the properties of hit-and-run sampling, it is shown that both the interior point method and the simulated annealing algorithm can solve a convex optimization problem in the membership oracle setting. The number of oracle calls made by these methods is bounded by a polynomial in the input size. The third method is an analytic center cutting plane method that shows promising performance for copositive optimization. It outperforms the first two methods by a significant margin on the problem of separating a matrix from the completely positive cone. The final method is based on Mosek’s algorithm for nonsymmetric conic optimization. With their scaling matrix, search direction, and neighborhood, we define a method that converges to a near-optimal solution in polynomial time
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