23,061 research outputs found
Asymptotic Cones of Quadratically Defined Sets and Their Applications to QCQPs
Quadratically constrained quadratic programs (QCQPs) are a set of optimization problems defined by a quadratic objective function and quadratic constraints. QCQPs cover a diverse set of problems, but the nonconvexity and unboundedness of quadratic constraints lead to difficulties in globally solving a QCQP. This thesis covers properties of unbounded quadratic constraints via a description of the asymptotic cone of a set defined by a single quadratic constraint. A description of the asymptotic cone is provided, including properties such as retractiveness and horizon directions.
Using the characterization of the asymptotic cone, we generalize existing results for bounded quadratically defined regions with non-intersecting constraints. The newer result provides a sufficient condition for when the intersection of the lifted convex hulls of quadratically defined sets equals the lifted convex hull of the intersection. This document goes further by expanding the non-intersecting property to cover affine linear constraints.
The Frank-Wolfe theorem provides conditions for when a problem defined by a quadratic objective function over affine linear constraints has an optimal solution. Over time, this theorem has been extended to cover cases involving convex quadratic constraints. We discuss more current results through the lens of the asymptotic cone of a quadratically defined set. This discussion expands current results and provides a sufficient condition for when a QCQP with one quadratic constraint with an indefinite Hessian has an optimal solution
Minimum L1-distance projection onto the boundary of a convex set: Simple characterization
We show that the minimum distance projection in the L1-norm from an interior
point onto the boundary of a convex set is achieved by a single, unidimensional
projection. Application of this characterization when the convex set is a
polyhedron leads to either an elementary minmax problem or a set of easily
solved linear programs, depending upon whether the polyhedron is given as the
intersection of a set of half spaces or as the convex hull of a set of extreme
points. The outcome is an easier and more straightforward derivation of the
special case results given in a recent paper by Briec.Comment: 5 page
A new look at nonnegativity on closed sets and polynomial optimization
We first show that a continuous function f is nonnegative on a closed set
if and only if (countably many) moment matrices of some signed
measure with support equal to K, are all positive semidefinite
(if is compact is an arbitrary finite Borel measure with support
equal to K. In particular, we obtain a convergent explicit hierarchy of
semidefinite (outer) approximations with {\it no} lifting, of the cone of
nonnegative polynomials of degree at most . Wen used in polynomial
optimization on certain simple closed sets \K (like e.g., the whole space
, the positive orthant, a box, a simplex, or the vertices of the
hypercube), it provides a nonincreasing sequence of upper bounds which
converges to the global minimum by solving a hierarchy of semidefinite programs
with only one variable. This convergent sequence of upper bounds complements
the convergent sequence of lower bounds obtained by solving a hierarchy of
semidefinite relaxations
On Minimal Valid Inequalities for Mixed Integer Conic Programs
We study disjunctive conic sets involving a general regular (closed, convex,
full dimensional, and pointed) cone K such as the nonnegative orthant, the
Lorentz cone or the positive semidefinite cone. In a unified framework, we
introduce K-minimal inequalities and show that under mild assumptions, these
inequalities together with the trivial cone-implied inequalities are sufficient
to describe the convex hull. We study the properties of K-minimal inequalities
by establishing algebraic necessary conditions for an inequality to be
K-minimal. This characterization leads to a broader algebraically defined class
of K- sublinear inequalities. We establish a close connection between
K-sublinear inequalities and the support functions of sets with a particular
structure. This connection results in practical ways of showing that a given
inequality is K-sublinear and K-minimal.
Our framework generalizes some of the results from the mixed integer linear
case. It is well known that the minimal inequalities for mixed integer linear
programs are generated by sublinear (positively homogeneous, subadditive and
convex) functions that are also piecewise linear. This result is easily
recovered by our analysis. Whenever possible we highlight the connections to
the existing literature. However, our study unveils that such a cut generating
function view treating the data associated with each individual variable
independently is not possible in the case of general cones other than
nonnegative orthant, even when the cone involved is the Lorentz cone
Preferences Yielding the "Precautionary Effect"
Consider an agent taking two successive decisions to maximize his expected
utility under uncertainty. After his first decision, a signal is revealed that
provides information about the state of nature. The observation of the signal
allows the decision-maker to revise his prior and the second decision is taken
accordingly. Assuming that the first decision is a scalar representing
consumption, the \emph{precautionary effect} holds when initial consumption is
less in the prospect of future information than without (no signal).
\citeauthor{Epstein1980:decision} in \citep*{Epstein1980:decision} has provided
the most operative tool to exhibit the precautionary effect. Epstein's Theorem
holds true when the difference of two convex functions is either convex or
concave, which is not a straightforward property, and which is difficult to
connect to the primitives of the economic model. Our main contribution consists
in giving a geometric characterization of when the difference of two convex
functions is convex, then in relating this to the primitive utility model. With
this tool, we are able to study and unite a large body of the literature on the
precautionary effect
Robust distributed linear programming
This paper presents a robust, distributed algorithm to solve general linear
programs. The algorithm design builds on the characterization of the solutions
of the linear program as saddle points of a modified Lagrangian function. We
show that the resulting continuous-time saddle-point algorithm is provably
correct but, in general, not distributed because of a global parameter
associated with the nonsmooth exact penalty function employed to encode the
inequality constraints of the linear program. This motivates the design of a
discontinuous saddle-point dynamics that, while enjoying the same convergence
guarantees, is fully distributed and scalable with the dimension of the
solution vector. We also characterize the robustness against disturbances and
link failures of the proposed dynamics. Specifically, we show that it is
integral-input-to-state stable but not input-to-state stable. The latter fact
is a consequence of a more general result, that we also establish, which states
that no algorithmic solution for linear programming is input-to-state stable
when uncertainty in the problem data affects the dynamics as a disturbance. Our
results allow us to establish the resilience of the proposed distributed
dynamics to disturbances of finite variation and recurrently disconnected
communication among the agents. Simulations in an optimal control application
illustrate the results
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