211 research outputs found
Characterizing the universal rigidity of generic frameworks
A framework is a graph and a map from its vertices to E^d (for some d). A
framework is universally rigid if any framework in any dimension with the same
graph and edge lengths is a Euclidean image of it. We show that a generic
universally rigid framework has a positive semi-definite stress matrix of
maximal rank. Connelly showed that the existence of such a positive
semi-definite stress matrix is sufficient for universal rigidity, so this
provides a characterization of universal rigidity for generic frameworks. We
also extend our argument to give a new result on the genericity of strict
complementarity in semidefinite programming.Comment: 18 pages, v2: updates throughout; v3: published versio
On the solution existence and stability of polynomial optimization problems
This paper introduces and investigates a regularity condition in the
asymptotic sense for the optimization problems whose objective functions are
polynomial. We prove two sufficient conditions for the existence of solutions
for polynomial optimization problems. Further, when the constraint sets are
semi-algebraic, we show results on the stability of the solution map of
polynomial optimization problems. At the end of the paper, we discuss the
genericity of the regularity condition.Comment: The old title: A regularity condition in polynomial optimizatio
SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization
In this paper, we introduce a new class of nonsmooth convex functions called
SOS-convex semialgebraic functions extending the recently proposed notion of
SOS-convex polynomials. This class of nonsmooth convex functions covers many
common nonsmooth functions arising in the applications such as the Euclidean
norm, the maximum eigenvalue function and the least squares functions with
-regularization or elastic net regularization used in statistics and
compressed sensing. We show that, under commonly used strict feasibility
conditions, the optimal value and an optimal solution of SOS-convex
semi-algebraic programs can be found by solving a single semi-definite
programming problem (SDP). We achieve the results by using tools from
semi-algebraic geometry, convex-concave minimax theorem and a recently
established Jensen inequality type result for SOS-convex polynomials. As an
application, we outline how the derived results can be applied to show that
robust SOS-convex optimization problems under restricted spectrahedron data
uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP
relaxation result for restricted ellipsoidal data uncertainty and answers the
open questions left in [Optimization Letters 9, 1-18(2015)] on how to recover a
robust solution from the semi-definite programming relaxation in this broader
setting
Formalized proof, computation, and the construction problem in algebraic geometry
An informal discussion of how the construction problem in algebraic geometry
motivates the search for formal proof methods. Also includes a brief discussion
of my own progress up to now, which concerns the formalization of category
theory within a ZFC-like environment
Generic optimality conditions for semi-algebraic convex programs
We consider linear optimization over a nonempty convex semi-algebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique \active" manifold, around which F is \partly smooth", and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F, and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F
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