3,673 research outputs found
Computing semiparametric bounds on the expected payments of insurance instruments via column generation
It has been recently shown that numerical semiparametric bounds on the
expected payoff of fi- nancial or actuarial instruments can be computed using
semidefinite programming. However, this approach has practical limitations.
Here we use column generation, a classical optimization technique, to address
these limitations. From column generation, it follows that practical univari-
ate semiparametric bounds can be found by solving a series of linear programs.
In addition to moment information, the column generation approach allows the
inclusion of extra information about the random variable; for instance,
unimodality and continuity, as well as the construction of corresponding
worst/best-case distributions in a simple way
Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations
It is well-known that any sum of squares (SOS) program can be cast as a
semidefinite program (SDP) of a particular structure and that therein lies the
computational bottleneck for SOS programs, as the SDPs generated by this
procedure are large and costly to solve when the polynomials involved in the
SOS programs have a large number of variables and degree. In this paper, we
review SOS optimization techniques and present two new methods for improving
their computational efficiency. The first method leverages the sparsity of the
underlying SDP to obtain computational speed-ups. Further improvements can be
obtained if the coefficients of the polynomials that describe the problem have
a particular sparsity pattern, called chordal sparsity. The second method
bypasses semidefinite programming altogether and relies instead on solving a
sequence of more tractable convex programs, namely linear and second order cone
programs. This opens up the question as to how well one can approximate the
cone of SOS polynomials by second order representable cones. In the last part
of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201
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
Approximation Limits of Linear Programs (Beyond Hierarchies)
We develop a framework for approximation limits of polynomial-size linear
programs from lower bounds on the nonnegative ranks of suitably defined
matrices. This framework yields unconditional impossibility results that are
applicable to any linear program as opposed to only programs generated by
hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations
for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound
applies to linear programs using a certain encoding of CLIQUE as a linear
optimization problem.) Moreover, we establish a similar result for
approximations of semidefinite programs by linear programs. Our main ingredient
is a quantitative improvement of Razborov's rectangle corruption lemma for the
high error regime, which gives strong lower bounds on the nonnegative rank of
certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure
On the existence of 0/1 polytopes with high semidefinite extension complexity
In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope
whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope
projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension
complexity is exponential. The question whether there exists a 0/1 polytope
with high PSD extension complexity was left open. We answer this question in
the affirmative by showing that there is a 0/1 polytope such that any
spectrahedron projecting to it must be the intersection of a semidefinite cone
of dimension~2^{\Omega(n)} and an affine space. Our proof relies on a new
technique to rescale semidefinite factorizations
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