3,255 research outputs found
On the convex hull of convex quadratic optimization problems with indicators
We consider the convex quadratic optimization problem with indicator
variables and arbitrary constraints on the indicators. We show that a convex
hull description of the associated mixed-integer set in an extended space with
a quadratic number of additional variables consists of a single positive
semidefinite constraint (explicitly stated) and linear constraints. In
particular, convexification of this class of problems reduces to describing a
polyhedral set in an extended formulation. While the vertex representation of
this polyhedral set is exponential and an explicit linear inequality
description may not be readily available in general, we derive a compact
mixed-integer linear formulation whose solutions coincide with the vertices of
the polyhedral set. We also give descriptions in the original space of
variables: we provide a description based on an infinite number of
conic-quadratic inequalities, which are ``finitely generated." In particular,
it is possible to characterize whether a given inequality is necessary to
describe the convex hull. The new theory presented here unifies several
previously established results, and paves the way toward utilizing polyhedral
methods to analyze the convex hull of mixed-integer nonlinear sets
Constrained Optimization of Rank-One Functions with Indicator Variables
Optimization problems involving minimization of a rank-one convex function
over constraints modeling restrictions on the support of the decision variables
emerge in various machine learning applications. These problems are often
modeled with indicator variables for identifying the support of the continuous
variables. In this paper we investigate compact extended formulations for such
problems through perspective reformulation techniques. In contrast to the
majority of previous work that relies on support function arguments and
disjunctive programming techniques to provide convex hull results, we propose a
constructive approach that exploits a hidden conic structure induced by
perspective functions. To this end, we first establish a convex hull result for
a general conic mixed-binary set in which each conic constraint involves a
linear function of independent continuous variables and a set of binary
variables. We then demonstrate that extended representations of sets associated
with epigraphs of rank-one convex functions over constraints modeling indicator
relations naturally admit such a conic representation. This enables us to
systematically give perspective formulations for the convex hull descriptions
of these sets with nonlinear separable or non-separable objective functions,
sign constraints on continuous variables, and combinatorial constraints on
indicator variables. We illustrate the efficacy of our results on sparse
nonnegative logistic regression problems
Compact extended formulations for low-rank functions with indicator variables
We study the mixed-integer epigraph of a low-rank convex function with
non-convex indicator constraints, which are often used to impose logical
constraints on the support of the solutions. Extended formulations describing
the convex hull of such sets can easily be constructed via disjunctive
programming, although a direct application of this method often yields
prohibitively large formulations, whose size is exponential in the number of
variables. In this paper, we propose a new disjunctive representation of the
sets under study, which leads to compact formulations with size exponential in
the rank of the function, but polynomial in the number of variables. Moreover,
we show how to project out the additional variables for the case of rank-one
functions, recovering or generalizing known results for the convex hulls of
such sets (in the original space of variables)
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A review of portfolio planning: Models and systems
In this chapter, we first provide an overview of a number of portfolio planning models
which have been proposed and investigated over the last forty years. We revisit the
mean-variance (M-V) model of Markowitz and the construction of the risk-return
efficient frontier. A piecewise linear approximation of the problem through a
reformulation involving diagonalisation of the quadratic form into a variable
separable function is also considered. A few other models, such as, the Mean
Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the
Minimax (MM) model which use alternative metrics for risk are also introduced,
compared and contrasted. Recently asymmetric measures of risk have gained in
importance; we consider a generic representation and a number of alternative
symmetric and asymmetric measures of risk which find use in the evaluation of
portfolios. There are a number of modelling and computational considerations which
have been introduced into practical portfolio planning problems. These include: (a)
buy-in thresholds for assets, (b) restriction on the number of assets (cardinality
constraints), (c) transaction roundlot restrictions. Practical portfolio models may also
include (d) dedication of cashflow streams, and, (e) immunization which involves
duration matching and convexity constraints. The modelling issues in respect of these
features are discussed. Many of these features lead to discrete restrictions involving
zero-one and general integer variables which make the resulting model a quadratic
mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the
algorithms and solution methods for this class of problems are also discussed. The
issues of preparing the analytic data (financial datamarts) for this family of portfolio
planning problems are examined. We finally present computational results which
provide some indication of the state-of-the-art in the solution of portfolio optimisation
problems
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