3,406 research outputs found
On Quadratic Programming with a Ratio Objective
Quadratic Programming (QP) is the well-studied problem of maximizing over
{-1,1} values the quadratic form \sum_{i \ne j} a_{ij} x_i x_j. QP captures
many known combinatorial optimization problems, and assuming the unique games
conjecture, semidefinite programming techniques give optimal approximation
algorithms. We extend this body of work by initiating the study of Quadratic
Programming problems where the variables take values in the domain {-1,0,1}.
The specific problems we study are
QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum
x_i^2}, and Normalized QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j}
a_{ij} x_i x_j}{\sum d_i x_i^2}, where d_i = \sum_j |a_{ij}|
We consider an SDP relaxation obtained by adding constraints to the natural
eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an
algorithm for QP-ratio. We also obtain an
approximation for bipartite graphs, and better algorithms
for special cases. As with other problems with ratio objectives (e.g. uniform
sparsest cut), it seems difficult to obtain inapproximability results based on
P!=NP. We give two results that indicate that QP-Ratio is hard to approximate
to within any constant factor. We also give a natural distribution on instances
of QP-Ratio for which an n^\epsilon approximation (for \epsilon roughly 1/10)
seems out of reach of current techniques
Towards a Resolution of P = NP Conjecture
In this research paper, the problem of optimization of a quadratic form over
the convex hull generated by the corners of hypercube is attempted and solved.
It is reasoned that under some conditions, the optimum occurs at the corners of
hypercube. Results related to the computation of global optimum stable state
(an NP hard problem) are discussed. An algorithm is proposed. It is hoped that
the results shed light on resolving the P not equal to NP problem.Comment: 15 pages. arXiv admin note: substantial text overlap with
arXiv:1207.063
Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs
We describe an efficient approximation algorithm for evaluating the
ground-state energy of the classical Ising Hamiltonian with linear terms on an
arbitrary planar graph. The running time of the algorithm grows linearly with
the number of spins and exponentially with 1/epsilon, where epsilon is the
worst-case relative error. This result contrasts the well known fact that exact
computation of the ground-state energy for the two-dimensional Ising spin glass
model is NP-hard. We also present a classical approximation algorithm for the
Local Hamiltonian Problem or Quantum Ising Spin Glass problem on a planar graph
with bounded degree which is known to be a QMA-complete problem. Using a
different technique we find a classical approximation algorithm for the quantum
Ising spin glass problem on the simplest planar graph with unbounded degree,
the star graph.Comment: 7 pages; v2 has some small corrections; the presentation in v3 has
been substantially revised. v4 is considerably expanded and includes our
results on quantum Ising spin glasse
A Study of Piecewise Linear-Quadratic Programs
Motivated by a growing list of nontraditional statistical estimation problems
of the piecewise kind, this paper provides a survey of known results
supplemented with new results for the class of piecewise linear-quadratic
programs. These are linearly constrained optimization problems with piecewise
linear-quadratic (PLQ) objective functions. Starting from a study of the
representation of such a function in terms of a family of elementary functions
consisting of squared affine functions, squared plus-composite-affine
functions, and affine functions themselves, we summarize some local properties
of a PLQ function in terms of their first and second-order directional
derivatives. We extend some well-known necessary and sufficient second-order
conditions for local optimality of a quadratic program to a PLQ program and
provide a dozen such equivalent conditions for strong, strict, and isolated
local optimality, showing in particular that a PLQ program has the same
characterizations for local minimality as a standard quadratic program. As a
consequence of one such condition, we show that the number of strong, strict,
or isolated local minima of a PLQ program is finite; this result supplements a
recent result about the finite number of directional stationary objective
values. Interestingly, these finiteness results can be uncovered by invoking a
very powerful property of subanalytic functions; our proof is fairly
elementary, however. We discuss applications of PLQ programs in some modern
statistical estimation problems. These problems lead to a special class of
unconstrained composite programs involving the non-differentiable
-function, for which we show that the task of verifying the
second-order stationary condition can be converted to the problem of checking
the copositivity of certain Schur complement on the nonnegative orthant
On the Worst-Case Approximability of Sparse PCA
It is well known that Sparse PCA (Sparse Principal Component Analysis) is
NP-hard to solve exactly on worst-case instances. What is the complexity of
solving Sparse PCA approximately? Our contributions include: 1) a simple and
efficient algorithm that achieves an -approximation; 2) NP-hardness
of approximation to within , for some small constant
; 3) SSE-hardness of approximation to within any constant
factor; and 4) an
("quasi-quasi-polynomial") gap for the standard semidefinite program.Comment: 20 page
ADMM for the SDP relaxation of the QAP
The semidefinite programming (SDP) relaxation has proven to be extremely
strong for many hard discrete optimization problems. This is in particular true
for the quadratic assignment problem (QAP), arguably one of the hardest NP-hard
discrete optimization problems. There are several difficulties that arise in
efficiently solving the SDP relaxation, e.g.,~increased dimension; inefficiency
of the current primal-dual interior point solvers in terms of both time and
accuracy; and difficulty and high expense in adding cutting plane constraints.
We propose using the alternating direction method of multipliers (ADMM) to
solve the SDP relaxation. This first order approach allows for inexpensive
iterations, a method of cheaply obtaining low rank solutions, as well a trivial
way of adding cutting plane inequalities. When compared to current approaches
and current best available bounds we obtain remarkable robustness, efficiency
and improved bounds.Comment: 12 pages, 1 tabl
First-order Methods Almost Always Avoid Saddle Points
We establish that first-order methods avoid saddle points for almost all
initializations. Our results apply to a wide variety of first-order methods,
including gradient descent, block coordinate descent, mirror descent and
variants thereof. The connecting thread is that such algorithms can be studied
from a dynamical systems perspective in which appropriate instantiations of the
Stable Manifold Theorem allow for a global stability analysis. Thus, neither
access to second-order derivative information nor randomness beyond
initialization is necessary to provably avoid saddle points
Convexifiability of Continuous and Discrete Nonnegative Quadratic Programs for Gap-Free Duality
In this paper we show that a convexifiability property of nonconvex quadratic
programs with nonnegative variables and quadratic constraints guarantees zero
duality gap between the quadratic programs and their semi-Lagrangian duals.
More importantly, we establish that this convexifiability is hidden in classes
of nonnegative homogeneous quadratic programs and discrete quadratic programs,
such as mixed integer quadratic programs, revealing zero duality gaps. As an
application, we prove that robust counterparts of uncertain mixed integer
quadratic programs with objective data uncertainty enjoy zero duality gaps
under suitable conditions. Various sufficient conditions for convexifiability
are also given
Rounding Lasserre SDPs using column selection and spectrum-based approximation schemes for graph partitioning and Quadratic IPs
We present an approximation scheme for minimizing certain Quadratic Integer
Programming problems with positive semidefinite objective functions and global
linear constraints. This framework includes well known graph problems such as
Minimum graph bisection, Edge expansion, Sparsest Cut, and Small Set expansion,
as well as the Unique Games problem. These problems are notorious for the
existence of huge gaps between the known algorithmic results and NP-hardness
results. Our algorithm is based on rounding semidefinite programs from the
Lasserre hierarchy, and the analysis uses bounds for low-rank approximations of
a matrix in Frobenius norm using columns of the matrix.
For all the above graph problems, we give an algorithm running in time
with approximation ratio
, where is the 'th
smallest eigenvalue of the normalized graph Laplacian . In the
case of graph bisection and small set expansion, the number of vertices in the
cut is within lower-order terms of the stipulated bound. Our results imply
factor approximation in time where
is the number of eigenvalues of smaller than (for
variants of sparsest cut, also
suffices, and as is usually on interesting instances of
these problems, this requirement on is typically weaker). For Unique
Games, we give a factor approximation for
minimizing the number of unsatisfied constraints in time,
improving upon an earlier bound for solving Unique Games on expanders. We also
give an algorithm for independent sets in graphs that performs well when the
Laplacian does not have too many eigenvalues bigger than .Comment: This manuscript is a merged and definitive version of (Guruswami,
Sinop: FOCS 2011) and (Guruswami, Sinop: SODA 2013), with a significantly
revised presentation. arXiv admin note: substantial text overlap with
arXiv:1104.474
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