4,172 research outputs found
Regularization vs. Relaxation: A conic optimization perspective of statistical variable selection
Variable selection is a fundamental task in statistical data analysis.
Sparsity-inducing regularization methods are a popular class of methods that
simultaneously perform variable selection and model estimation. The central
problem is a quadratic optimization problem with an l0-norm penalty. Exactly
enforcing the l0-norm penalty is computationally intractable for larger scale
problems, so dif- ferent sparsity-inducing penalty functions that approximate
the l0-norm have been introduced. In this paper, we show that viewing the
problem from a convex relaxation perspective offers new insights. In
particular, we show that a popular sparsity-inducing concave penalty function
known as the Minimax Concave Penalty (MCP), and the reverse Huber penalty
derived in a recent work by Pilanci, Wainwright and Ghaoui, can both be derived
as special cases of a lifted convex relaxation called the perspective
relaxation. The optimal perspective relaxation is a related minimax problem
that balances the overall convexity and tightness of approximation to the l0
norm. We show it can be solved by a semidefinite relaxation. Moreover, a
probabilistic interpretation of the semidefinite relaxation reveals connections
with the boolean quadric polytope in combinatorial optimization. Finally by
reformulating the l0-norm pe- nalized problem as a two-level problem, with the
inner level being a Max-Cut problem, our proposed semidefinite relaxation can
be realized by replacing the inner level problem with its semidefinite
relaxation studied by Goemans and Williamson. This interpretation suggests
using the Goemans-Williamson rounding procedure to find approximate solutions
to the l0-norm penalized problem. Numerical experiments demonstrate the
tightness of our proposed semidefinite relaxation, and the effectiveness of
finding approximate solutions by Goemans-Williamson rounding.Comment: Also available on optimization online
{http://www.optimization-online.org/DB_HTML/2015/05/4932.html
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
Semidefinite relaxation based branch-and-bound method for nonconvex quadratic programming
Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006.Includes bibliographical references (leaves 73-75).In this thesis, we use a semidefinite relaxation based branch-and-bound method to solve nonconvex quadratic programming problems. Firstly, we show an interval branch-and-bound method to calculate the bounds for the minimum of bounded polynomials. Then we demonstrate four SDP relaxation methods to solve nonconvex Box constrained Quadratic Programming (BoxQP) problems and the comparison of the four methods. For some lower dimensional problems, SDP relaxation methods can achieve tight bounds for the BoxQP problem; whereas for higher dimensional cases (more than 20 dimensions), the bounds achieved by the four Semidefinite programming (SDP) relaxation methods are always loose. To achieve tight bounds for higher dimensional BoxQP problems, we combine the branch-and-bound method and SDP relaxation method to develop an SDP relaxation based branch-and-bound (SDPBB) method. We introduce a sensitivity analysis method for the branching process of SDPBB. This sensitivity analysis method can improve the convergence speed significantly.(cont.) Compared to the interval branch-and-bound method and the global optimization software BARON, SDPBB can achieve better bounds and is also much more efficient. Additionally, we have developed a multisection algorithm for SDPBB and the multisection algorithm has been parallelized using Message Passing Interface (MPI). By parallelizing the program, we can significantly improve the speed of solving higher dimensional BoxQP problems.by Sha Hu.S.M
Concave Quadratic Cuts for Mixed-Integer Quadratic Problems
The technique of semidefinite programming (SDP) relaxation can be used to
obtain a nontrivial bound on the optimal value of a nonconvex quadratically
constrained quadratic program (QCQP). We explore concave quadratic inequalities
that hold for any vector in the integer lattice , and show that
adding these inequalities to a mixed-integer nonconvex QCQP can improve the
SDP-based bound on the optimal value. This scheme is tested using several
numerical problem instances of the max-cut problem and the integer least
squares problem.Comment: 24 pages, 1 figur
Penalized Semidefinite Programming for Quadratically-Constrained Quadratic Optimization
In this paper, we give a new penalized semidefinite programming approach for
non-convex quadratically-constrained quadratic programs (QCQPs). We incorporate
penalty terms into the objective of convex relaxations in order to retrieve
feasible and near-optimal solutions for non-convex QCQPs. We introduce a
generalized linear independence constraint qualification (GLICQ) criterion and
prove that any GLICQ regular point that is sufficiently close to the feasible
set can be used to construct an appropriate penalty term and recover a feasible
solution. Inspired by these results, we develop a heuristic sequential
procedure that preserves feasibility and aims to improve the objective value at
each iteration. Numerical experiments on large-scale system identification
problems as well as benchmark instances from the library of quadratic
programming (QPLIB) demonstrate the ability of the proposed penalized
semidefinite programs in finding near-optimal solutions for non-convex QCQP
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
We present algorithm MIQCR-CB that is an advancement of method
MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving
mixed-integer quadratic programs and works in two phases: the first phase
determines an equivalent quadratic formulation with a convex objective function
by solving a semidefinite problem , and, in the second phase, the
equivalent formulation is solved by a standard solver. As the reformulation
relies on the solution of a large-scale semidefinite program, it is not
tractable by existing semidefinite solvers, already for medium sized problems.
To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm
within a Lagrangian duality framework for solving that substantially
speeds up the first phase. Moreover, this algorithm leads to a reformulated
problem of smaller size than the one obtained by the original MIQCR method
which results in a shorter time for solving the second phase.
We present extensive computational results to show the efficiency of our
algorithm
Conic relaxation approaches for equal deployment problems
An important problem in the breeding of livestock, crops, and forest trees is
the optimum of selection of genotypes that maximizes genetic gain. The key
constraint in the optimal selection is a convex quadratic constraint that
ensures genetic diversity, therefore, the optimal selection can be cast as a
second-order cone programming (SOCP) problem. Yamashita et al. (2015) exploits
the structural sparsity of the quadratic constraints and reduces the
computation time drastically while attaining the same optimal solution.
This paper is concerned with the special case of equal deployment (ED), in
which we solve the optimal selection problem with the constraint that
contribution of genotypes must either be a fixed size or zero. This involves a
nature of combinatorial optimization, and the ED problem can be described as a
mixed-integer SOCP problem.
In this paper, we discuss conic relaxation approaches for the ED problem
based on LP (linear programming), SOCP, and SDP (semidefinite programming). We
analyze theoretical bounds derivedfrom the SDP relaxation approaches using the
work of Tseng (2003) and show that the theoretical bounds are not quite sharp
for tree breeding problems. We propose a steepest-ascent method that combines
the solution obtained from the conic relaxation problems with a concept from
discrete convex optimization in order to acquire an approximate solution for
the ED problem in a practical time. From numerical tests, we observed that
among the LP, SOCP, and SDP relaxation problems, SOCP gave a suitable solution
from the viewpoints of the optimal values and the computation time. The
steepest-ascent method starting from the SOCP solution provides high-quality
solutions much faster than an existing method that has been widely used for the
optimal selection problems and a branch-and-bound method
A Scalable Semidefinite Relaxation Approach to Grid Scheduling
Determination of the most economic strategies for supply and transmission of
electricity is a daunting computational challenge. Due to theoretical barriers,
so-called NP-hardness, the amount of effort to optimize the schedule of
generating units and route of power, can grow exponentially with the number of
decision variables. Practical approaches to this problem involve legacy
approximations and ad-hoc heuristics that may undermine the efficiency and
reliability of power system operations, that are ever growing in scale and
complexity. Therefore, the development of powerful optimization methods for
detailed power system scheduling is critical to the realization of smart grids
and has received significant attention recently. In this paper, we propose for
the first time a computational method, which is capable of solving large-scale
power system scheduling problems with thousands of generating units, while
accurately incorporating the nonlinear equations that govern the flow of
electricity on the grid. The utilization of this accurate nonlinear model, as
opposed to its linear approximations, results in a more efficient and
transparent market design, as well as improvements in the reliability of power
system operations. We design a polynomial-time solvable third-order
semidefinite programming (TSDP) relaxation, with the aim of finding a near
globally optimal solution for the original NP-hard problem. The proposed method
is demonstrated on the largest available benchmark instances from real-world
European grid data, for which provably optimal or near-optimal solutions are
obtained
A Semidefinite Relaxation for Air Traffic Flow Scheduling
We first formulate the problem of optimally scheduling air traffic low with
sector capacity constraints as a mixed integer linear program. We then use
semidefinite relaxation techniques to form a convex relaxation of that problem.
Finally, we present a randomization algorithm to further improve the quality of
the solution. Because of the specific structure of the air traffic flow
problem, the relaxation has a single semidefinite constraint of size dn where d
is the maximum delay and n the number of flights.Comment: Submitted to RIVF 200
Solution to the Inverse Wulff Problem by Means of the Enhanced Semidefinite Relaxation Method
We propose a novel method of resolving the optimal anisotropy function. The
idea is to construct the optimal anisotropy function as a solution to the
inverse Wulff problem, i.e. as a minimizer for the anisoperimetric ratio for a
given Jordan curve in the plane. It leads to a nonconvex quadratic optimization
problem with linear matrix inequalities. In order to solve it we propose the
so-called enhanced semidefinite relaxation method which is based on a solution
to a convex semidefinite problem obtained by a semidefinite relaxation of the
original problem augmented by quadratic-linear constraints. We show that the
sequence of finite dimensional approximations of the optimal anisoperimetric
ratio converges to the optimal anisoperimetric ratio which is a solution to the
inverse Wulff problem. Several computational examples, including those
corresponding to boundaries of real snowflakes and discussion on the rate of
convergence of numerical method are also presented in this paper.Comment: 5 figure
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