11 research outputs found
Optimality certificates for convex minimization and Helly numbers
We consider the problem of minimizing a convex function over a subset of R^n
that is not necessarily convex (minimization of a convex function over the
integer points in a polytope is a special case). We define a family of duals
for this problem and show that, under some natural conditions, strong duality
holds for a dual problem in this family that is more restrictive than
previously considered duals.Comment: 5 page
Duality for mixed-integer convex minimization
We extend in two ways the standard Karush–Kuhn–Tucker optimality conditions to problems with a convex objective, convex functional constraints, and the extra requirement that some of the variables must be integral. While the standard Karush–Kuhn–Tucker conditions involve separating hyperplanes, our extension is based on mixed-integer-free polyhedra. Our optimality conditions allow us to define an exact dual of our original mixed-integer convex problem
On Subadditive Duality for Conic Mixed-Integer Programs
In this paper, we show that the subadditive dual of a feasible conic
mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover,
we show that this dual feasibility condition is equivalent to feasibility of
the conic dual of the continuous relaxation of the conic MIP. In addition, we
prove that all known conditions and other 'natural' conditions for strong
duality, such as strict mixed-integer feasibility, boundedness of the feasible
set or essentially strict feasibility imply that the subadditive dual is
feasible. As an intermediate result, we extend the so-called 'finiteness
property' from full-dimensional convex sets to intersections of
full-dimensional convex sets and Dirichlet convex sets
On subadditive duality for conic mixed-integer programs
In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible. Moreover, we show that this dual feasibility condition is equivalent to feasibility of the conic dual of the continuous relaxation of the conic MIP. In addition, we prove that all known conditions and other 'natural' conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set or essentially strict feasibility imply that the subadditive dual is feasible. As an intermediate result, we extend the so-called 'finiteness property' from full-dimensional convex sets to intersections of full-dimensional convex sets and Dirichlet convex sets
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
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CROSS: a framework for cyber risk optimisation in smart homes
This work introduces a decision support framework, called Cyber Risk Optimiser for Smart homeS (CROSS), which advises both smart home users and smart home service providers on how to select an optimal portfolio of cyber security controls to counteract cyber attacks in a smart home including traditional cyber attacks and adversarial machine learning attacks. CROSS is based on a multi-objective bi-level two-stage optimisation. In stage-one optimisation, the problem is modelled as a multi-leader-follower game that considers both security and economic objectives, where the provider selects a security portfolio to protect both itself and its users, while rational attackers target the weakest path. Stage-two optimisation is a Stackelberg security game that focuses on additional user security controls under the remit of smart home users. While CROSS can potentially be applied to other similar use cases, in this paper, our aim is to address threats against artificial intelligence (AI) applications as the use of AI in smart Internet of Things (IoT) devices introduces new cyber threats to home environments. Specifically, we have implemented and assessed CROSS in a smart heating use case in a prototypical AI-enabled IoT environment that combines characteristics and vulnerabilities currently present on existing commercial off-the-shelf (COTS) devices, demonstrating the selection of optimal decisions