35,763 research outputs found
The proximal distance algorithm
The MM principle is a device for creating optimization algorithms satisfying
the ascent or descent property. The current survey emphasizes the role of the
MM principle in nonlinear programming. For smooth functions, one can construct
an adaptive interior point method based on scaled Bregmann barriers. This
algorithm does not follow the central path. For convex programming subject to
nonsmooth constraints, one can combine an exact penalty method with distance
majorization to create versatile algorithms that are effective even in discrete
optimization. These proximal distance algorithms are highly modular and reduce
to set projections and proximal mappings, both very well-understood techniques
in optimization. We illustrate the possibilities in linear programming, binary
piecewise-linear programming, nonnegative quadratic programming,
regression, matrix completion, and inverse sparse covariance estimation.Comment: 22 pages, 0 figures, 8 tables, modified from conference publicatio
Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra
We consider minimizing a conic quadratic objective over a polyhedron. Such
problems arise in parametric value-at-risk minimization, portfolio
optimization, and robust optimization with ellipsoidal objective uncertainty;
and they can be solved by polynomial interior point algorithms for conic
quadratic optimization. However, interior point algorithms are not well-suited
for branch-and-bound algorithms for the discrete counterparts of these problems
due to the lack of effective warm starts necessary for the efficient solution
of convex relaxations repeatedly at the nodes of the search tree.
In order to overcome this shortcoming, we reformulate the problem using the
perspective of the quadratic function. The perspective reformulation lends
itself to simple coordinate descent and bisection algorithms utilizing the
simplex method for quadratic programming, which makes the solution methods
amenable to warm starts and suitable for branch-and-bound algorithms. We test
the simplex-based quadratic programming algorithms to solve convex as well as
discrete instances and compare them with the state-of-the-art approaches. The
computational experiments indicate that the proposed algorithms scale much
better than interior point algorithms and return higher precision solutions. In
our experiments, for large convex instances, they provide up to 22x speed-up.
For smaller discrete instances, the speed-up is about 13x over a barrier-based
branch-and-bound algorithm and 6x over the LP-based branch-and-bound algorithm
with extended formulations
A Second-Order Cone Based Approach for Solving the Trust Region Subproblem and Its Variants
We study the trust-region subproblem (TRS) of minimizing a nonconvex
quadratic function over the unit ball with additional conic constraints.
Despite having a nonconvex objective, it is known that the classical TRS and a
number of its variants are polynomial-time solvable. In this paper, we follow a
second-order cone (SOC) based approach to derive an exact convex reformulation
of the TRS under a structural condition on the conic constraint. Our structural
condition is immediately satisfied when there is no additional conic
constraints, and it generalizes several such conditions studied in the
literature. As a result, our study highlights an explicit connection between
the classical nonconvex TRS and smooth convex quadratic minimization, which
allows for the application of cheap iterative methods such as Nesterov's
accelerated gradient descent, to the TRS. Furthermore, under slightly stronger
conditions, we give a low-complexity characterization of the convex hull of the
epigraph of the nonconvex quadratic function intersected with the constraints
defining the domain without any additional variables. We also explore the
inclusion of additional hollow constraints to the domain of the TRS, and
convexification of the associated epigraph
Proximal algorithms for constrained composite optimization, with applications to solving low-rank SDPs
We study a family of (potentially non-convex) constrained optimization
problems with convex composite structure. Through a novel analysis of
non-smooth geometry, we show that proximal-type algorithms applied to exact
penalty formulations of such problems exhibit local linear convergence under a
quadratic growth condition, which the compositional structure we consider
ensures. The main application of our results is to low-rank semidefinite
optimization with Burer-Monteiro factorizations. We precisely identify the
conditions for quadratic growth in the factorized problem via structures in the
semidefinite problem, which could be of independent interest for understanding
matrix factorization
DC Decomposition of Nonconvex Polynomials with Algebraic Techniques
We consider the problem of decomposing a multivariate polynomial as the
difference of two convex polynomials. We introduce algebraic techniques which
reduce this task to linear, second order cone, and semidefinite programming.
This allows us to optimize over subsets of valid difference of convex
decompositions (dcds) and find ones that speed up the convex-concave procedure
(CCP). We prove, however, that optimizing over the entire set of dcds is
NP-hard
Proximal Distance Algorithms: Theory and Examples
Proximal distance algorithms combine the classical penalty method of
constrained minimization with distance majorization. If is
the loss function, and is the constraint set in a constrained minimization
problem, then the proximal distance principle mandates minimizing the penalized
loss and following the
solution to its limit as tends to . At
each iteration the squared Euclidean distance
is majorized by the spherical quadratic , where
denotes the projection of the current iterate onto . The
minimum of the surrogate function
is
given by the proximal map .
The next iterate automatically decreases the original
penalized loss for fixed . Since many explicit projections and proximal
maps are known, it is straightforward to derive and implement novel
optimization algorithms in this setting. These algorithms can take hundreds if
not thousands of iterations to converge, but the stereotyped nature of each
iteration makes proximal distance algorithms competitive with traditional
algorithms. For convex problems, we prove global convergence. Our numerical
examples include a) linear programming, b) nonnegative quadratic programming,
c) projection to the closest kinship matrix, d) projection onto a second-order
cone constraint, e) calculation of Horn's copositive matrix index, f) linear
complementarity programming, and g) sparse principal components analysis. The
proximal distance algorithm in each case is competitive or superior in speed to
traditional methods.Comment: 23 pages, 2 figures, 7 table
Approximate global minimizers to pairwise interaction problems via convex relaxation
We present a new approach for computing approximate global minimizers to a
large class of non-local pairwise interaction problems defined over probability
distributions. The approach predicts candidate global minimizers, with a
recovery guarantee, that are sometimes exact, and often within a few percent of
the optimum energy (under appropriate normalization of the energy). The
procedure relies on a convex relaxation of the pairwise energy that exploits
translational symmetry, followed by a recovery procedure that minimizes a
relative entropy. Numerical discretizations of the convex relaxation yield a
linear programming problem over convex cones that can be solved using
well-known methods. One advantage of the approach is that it provides
sufficient conditions for global minimizers to a non-convex quadratic
variational problem, in the form of a linear, convex, optimization problem for
the auto-correlation of the probability density. We demonstrate the approach in
a periodic domain for examples arising from models in materials, social
phenomena and flocking. The approach also exactly recovers the global minimizer
when a lattice of Dirac masses solves the convex relaxation. An important
by-product of the relaxation is a decomposition of the pairwise energy
functional into the sum of a convex functional and non-convex functional. We
observe that in some cases, the non-convex component of the decomposition can
be used to characterize the support of the recovered minimizers.Comment: 43 pages, 12 figure
Stochastic Control with Affine Dynamics and Extended Quadratic Costs
An extended quadratic function is a quadratic function plus the indicator
function of an affine set, that is, a quadratic function with embedded linear
equality constraints. We show that, under some technical conditions, random
convex extended quadratic functions are closed under addition, composition with
an affine function, expectation, and partial minimization, that is, minimizing
over some of its arguments. These properties imply that dynamic programming can
be tractably carried out for stochastic control problems with random affine
dynamics and extended quadratic cost functions. While the equations for the
dynamic programming iterations are much more complicated than for traditional
linear quadratic control, they are well suited to an object-oriented
implementation, which we describe. We also describe a number of known and new
applications.Comment: 46 pages, 16 figure
Path Following in the Exact Penalty Method of Convex Programming
Classical penalty methods solve a sequence of unconstrained problems that put
greater and greater stress on meeting the constraints. In the limit as the
penalty constant tends to , one recovers the constrained solution. In
the exact penalty method, squared penalties are replaced by absolute value
penalties, and the solution is recovered for a finite value of the penalty
constant. In practice, the kinks in the penalty and the unknown magnitude of
the penalty constant prevent wide application of the exact penalty method in
nonlinear programming. In this article, we examine a strategy of path following
consistent with the exact penalty method. Instead of performing optimization at
a single penalty constant, we trace the solution as a continuous function of
the penalty constant. Thus, path following starts at the unconstrained solution
and follows the solution path as the penalty constant increases. In the
process, the solution path hits, slides along, and exits from the various
constraints. For quadratic programming, the solution path is piecewise linear
and takes large jumps from constraint to constraint. For a general convex
program, the solution path is piecewise smooth, and path following operates by
numerically solving an ordinary differential equation segment by segment. Our
diverse applications to a) projection onto a convex set, b) nonnegative least
squares, c) quadratically constrained quadratic programming, d) geometric
programming, and e) semidefinite programming illustrate the mechanics and
potential of path following. The final detour to image denoising demonstrates
the relevance of path following to regularized estimation in inverse problems.
In regularized estimation, one follows the solution path as the penalty
constant decreases from a large value
Adaptive Restart for Accelerated Gradient Schemes
In this paper we demonstrate a simple heuristic adaptive restart technique
that can dramatically improve the convergence rate of accelerated gradient
schemes. The analysis of the technique relies on the observation that these
schemes exhibit two modes of behavior depending on how much momentum is
applied. In what we refer to as the 'high momentum' regime the iterates
generated by an accelerated gradient scheme exhibit a periodic behavior, where
the period is proportional to the square root of the local condition number of
the objective function. This suggests a restart technique whereby we reset the
momentum whenever we observe periodic behavior. We provide analysis to show
that in many cases adaptively restarting allows us to recover the optimal rate
of convergence with no prior knowledge of function parameters.Comment: 17 pages, 7 figure
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