13,624 research outputs found
The use of Grossone in Mathematical Programming and Operations Research
The concepts of infinity and infinitesimal in mathematics date back to
anciens Greek and have always attracted great attention. Very recently, a new
methodology has been proposed by Sergeyev for performing calculations with
infinite and infinitesimal quantities, by introducing an infinite unit of
measure expressed by the numeral grossone. An important characteristic of this
novel approach is its attention to numerical aspects. In this paper we will
present some possible applications and use of grossone in Operations Research
and Mathematical Programming. In particular, we will show how the use of
grossone can be beneficial in anti--cycling procedure for the well-known
simplex method for solving Linear Programming Problems and in defining exact
differentiable Penalty Functions in Nonlinear Programming
On the Burer-Monteiro method for general semidefinite programs
Consider a semidefinite program (SDP) involving an positive
semidefinite matrix . The Burer-Monteiro method uses the substitution to obtain a nonconvex optimization problem in terms of an
matrix . Boumal et al. showed that this nonconvex method provably solves
equality-constrained SDPs with a generic cost matrix when , where is the number of constraints. In this note we extend
their result to arbitrary SDPs, possibly involving inequalities or multiple
semidefinite constraints. We derive similar guarantees for a fixed cost matrix
and generic constraints. We illustrate applications to matrix sensing and
integer quadratic minimization.Comment: 10 page
Necessary Optimality Conditions for Continuous-Time Optimization Problems with Equality and Inequality Constraints
The paper is devoted to obtain first and second order necessary optimality
conditions for continuous-time optimization problems with equality and
inequality constraints. A full rank type regularity condition along with an
uniform implicit function theorem are used in order to establish such necessary
conditions.Comment: 20 page
A Method to Guarantee Local Convergence for Sequential Quadratic Programming with Poor Hessian Approximation
Sequential Quadratic Programming (SQP) is a powerful class of algorithms for
solving nonlinear optimization problems. Local convergence of SQP algorithms is
guaranteed when the Hessian approximation used in each Quadratic Programming
subproblem is close to the true Hessian. However, a good Hessian approximation
can be expensive to compute. Low cost Hessian approximations only guarantee
local convergence under some assumptions, which are not always satisfied in
practice. To address this problem, this paper proposes a simple method to
guarantee local convergence for SQP with poor Hessian approximation. The
effectiveness of the proposed algorithm is demonstrated in a numerical example
Constraint Qualifications and Optimality Conditions for Nonconvex Semi-Infinite and Infinite Programs
The paper concerns the study of new classes of nonlinear and nonconvex
optimization problems of the so-called infinite programming that are generally
defined on infinite-dimensional spaces of decision variables and contain
infinitely many of equality and inequality constraints with arbitrary (may not
be compact) index sets. These problems reduce to semi-infinite programs in the
case of finite-dimensional spaces of decision variables. We extend the
classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications
to such infinite and semi-infinite programs. The new qualification conditions
are used for efficient computing the appropriate normal cones to sets of
feasible solutions for these programs by employing advanced tools of
variational analysis and generalized differentiation. In the further
development we derive first-order necessary optimality conditions for infinite
and semi-infinite programs, which are new in both finite-dimensional and
infinite-dimensional settings.Comment: 28 page
A sequential semidefinite programming method and an application in passive reduced-order modeling
We consider the solution of nonlinear programs with nonlinear
semidefiniteness constraints. The need for an efficient exploitation of the
cone of positive semidefinite matrices makes the solution of such nonlinear
semidefinite programs more complicated than the solution of standard nonlinear
programs. In particular, a suitable symmetrization procedure needs to be chosen
for the linearization of the complementarity condition. The choice of the
symmetrization procedure can be shifted in a very natural way to certain linear
semidefinite subproblems, and can thus be reduced to a well-studied problem.
The resulting sequential semidefinite programming (SSP) method is a
generalization of the well-known SQP method for standard nonlinear programs. We
present a sensitivity result for nonlinear semidefinite programs, and then
based on this result, we give a self-contained proof of local quadratic
convergence of the SSP method. We also describe a class of nonlinear
semidefinite programs that arise in passive reduced-order modeling, and we
report results of some numerical experiments with the SSP method applied to
problems in that class
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