4,445 research outputs found
Approximate Convex Optimization by Online Game Playing
Lagrangian relaxation and approximate optimization algorithms have received
much attention in the last two decades. Typically, the running time of these
methods to obtain a approximate solution is proportional to
. Recently, Bienstock and Iyengar, following Nesterov,
gave an algorithm for fractional packing linear programs which runs in
iterations. The latter algorithm requires to solve a
convex quadratic program every iteration - an optimization subroutine which
dominates the theoretical running time.
We give an algorithm for convex programs with strictly convex constraints
which runs in time proportional to . The algorithm does NOT
require to solve any quadratic program, but uses gradient steps and elementary
operations only. Problems which have strictly convex constraints include
maximum entropy frequency estimation, portfolio optimization with loss risk
constraints, and various computational problems in signal processing.
As a side product, we also obtain a simpler version of Bienstock and
Iyengar's result for general linear programming, with similar running time.
We derive these algorithms using a new framework for deriving convex
optimization algorithms from online game playing algorithms, which may be of
independent interest
Globally Optimal Energy-Efficient Power Control and Receiver Design in Wireless Networks
The characterization of the global maximum of energy efficiency (EE) problems
in wireless networks is a challenging problem due to the non-convex nature of
investigated problems in interference channels. The aim of this work is to
develop a new and general framework to achieve globally optimal solutions.
First, the hidden monotonic structure of the most common EE maximization
problems is exploited jointly with fractional programming theory to obtain
globally optimal solutions with exponential complexity in the number of network
links. To overcome this issue, we also propose a framework to compute
suboptimal power control strategies characterized by affordable complexity.
This is achieved by merging fractional programming and sequential optimization.
The proposed monotonic framework is used to shed light on the ultimate
performance of wireless networks in terms of EE and also to benchmark the
performance of the lower-complexity framework based on sequential programming.
Numerical evidence is provided to show that the sequential fractional
programming framework achieves global optimality in several practical
communication scenarios.Comment: Accepted for publication in the IEEE Transactions on Signal
Processin
MM Algorithms for Geometric and Signomial Programming
This paper derives new algorithms for signomial programming, a generalization
of geometric programming. The algorithms are based on a generic principle for
optimization called the MM algorithm. In this setting, one can apply the
geometric-arithmetic mean inequality and a supporting hyperplane inequality to
create a surrogate function with parameters separated. Thus, unconstrained
signomial programming reduces to a sequence of one-dimensional minimization
problems. Simple examples demonstrate that the MM algorithm derived can
converge to a boundary point or to one point of a continuum of minimum points.
Conditions under which the minimum point is unique or occurs in the interior of
parameter space are proved for geometric programming. Convergence to an
interior point occurs at a linear rate. Finally, the MM framework easily
accommodates equality and inequality constraints of signomial type. For the
most important special case, constrained quadratic programming, the MM
algorithm involves very simple updates.Comment: 16 pages, 1 figur
A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks
Research Report UPC-DEIO DR 2018-01. November 2018The computation of the Newton direction is the most time consuming step of interior-point methods. This direction was efficiently computed by a combination
of Cholesky factorizations and conjugate gradients in a specialized interior-point method for block-angular structured problems. In this work we apply this algorithmic approach to solve very large instances of minimum cost flows problems in bipartite networks, for convex objective functions with diagonal Hessians (i.e., either linear, quadratic or separable nonlinear objectives). After analyzing the theoretical properties of the interior-point method for this kind of problems, we provide extensive computational experiments with linear and quadratic instances of up to one billion arcs and 200 and five million nodes in each subset of the node partition. For linear and quadratic instances our approach is compared with the barriers algorithms of CPLEX (both standard path-following and homogeneous-self-dual); for linear instances it is also compared with the different algorithms of the state-of-the-art network flow solver LEMON (namely: network simplex, capacity scaling, cost scaling and cycle canceling).
The specialized interior-point approach significantly outperformed the other approaches in most of the linear and quadratic transportation instances tested. In particular, it always provided a solution within the time limit and it never exhausted the 192 Gigabytes of memory of the server used for the runs. For assignment problems the network algorithms in LEMON were the most efficient option.Peer ReviewedPreprin
An SDP Approach For Solving Quadratic Fractional Programming Problems
This paper considers a fractional programming problem (P) which minimizes a
ratio of quadratic functions subject to a two-sided quadratic constraint. As is
well-known, the fractional objective function can be replaced by a parametric
family of quadratic functions, which makes (P) highly related to, but more
difficult than a single quadratic programming problem subject to a similar
constraint set. The task is to find the optimal parameter and then
look for the optimal solution if is attained. Contrasted with the
classical Dinkelbach method that iterates over the parameter, we propose a
suitable constraint qualification under which a new version of the S-lemma with
an equality can be proved so as to compute directly via an exact
SDP relaxation. When the constraint set of (P) is degenerated to become an
one-sided inequality, the same SDP approach can be applied to solve (P) {\it
without any condition}. We observe that the difference between a two-sided
problem and an one-sided problem lies in the fact that the S-lemma with an
equality does not have a natural Slater point to hold, which makes the former
essentially more difficult than the latter. This work does not, either, assume
the existence of a positive-definite linear combination of the quadratic terms
(also known as the dual Slater condition, or a positive-definite matrix
pencil), our result thus provides a novel extension to the so-called "hard
case" of the generalized trust region subproblem subject to the upper and the
lower level set of a quadratic function.Comment: 26 page
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