729 research outputs found
Valid Inequalities and Reformulation Techniques for Mixed Integer Nonlinear Programming
One of the most important breakthroughs in the area of Mixed Integer Linear Programming (MILP) is the characterization of the convex hull of specially structured non-convex polyhedral sets in order to develop valid inequalities or cutting planes. Development of strong valid inequalities such as Split cuts, Gomory Mixed Integer (GMI) cuts, and Mixed Integer Rounding (MIR) cuts has resulted in highly effective branch-and-cut algorithms. While such cuts are known to be equivalent, each of their characterizations provides different advantages and insights.
The study of cutting planes for Mixed Integer Nonlinear Programming (MINLP) is still much more limited than that for MILP, since characterizing cuts for MINLP requires the study of the convex hull of a non-convex and non-polyhedral set, which has proven to be significantly harder than the polyhedral case. However, there has been significant work on the computational use of cuts in MINLP. Furthermore, there has recently been a significant interest in extending the associated theoretical results from MILP to the realm of MINLP.
This dissertation is focused on the development of new cuts and extended formulations for Mixed Integer Nonlinear Programs. We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. We also study the relation between the introduced cuts and some known classes of cutting planes from MILP. Furthermore, we show how an aggregation technique can be easily extended to characterize the convex hull of sets defined by two quadratic or by a conic quadratic and a quadratic inequality. We also computationally evaluate the performance of the introduced cuts and extended formulations on two classes of MINLP problems
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
A characterization of maximal homogeneous-quadratic-free sets
The intersection cut framework was introduced by Balas in 1971 as a method
for generating cutting planes in integer optimization. In this framework, one
uses a full-dimensional convex -free set, where is the feasible region
of the integer program, to derive a cut separating from a non-integral
vertex of a linear relaxation of . Among all -free sets, it is the
inclusion-wise maximal ones that yield the strongest cuts. Recently, this
framework has been extended beyond the integer case in order to obtain cutting
planes in non-linear settings. In this work, we consider the specific setting
when is defined by a homogeneous quadratic inequality. In this
'quadratic-free' setting, every function , where is
the unit disk in , generates a representation of a quadratic-free
set. While not every generates a maximal quadratic free set, it is the
case that every full-dimensional maximal quadratic free set is generated by
some . Our main result shows that the corresponding quadratic-free set
is full-dimensional and maximal if and only if is non-expansive and
satisfies a technical condition. This result yields a broader class of maximal
-free sets than previously known. Our result stems from a new
characterization of maximal -free sets (for general beyond the quadratic
setting) based on sequences that 'expose' inequalities defining the -free
set
Mixed Integer Second Order Cone Optimization, Disjunctive Conic Cuts: Theory and experiments
Mixed Integer Second Order Cone Optimization (MISOCO) problems allow practitioners to mathematically describe a wide variety of real world engineering problems including supply chain, finance, and networks design. A MISOCO problem minimizes a linear function over the set of solutions of a system of linear equations and the Cartesian product of second order cones of various dimensions, where a subset of the variables is constrained to be integer. This thesis presents a technique to derive inequalities that help to obtain a tighter mathematical description of the feasible set of a MISOCO problem. This improved description of the problem usually leads to accelerate the process of finding its optimal solution. In this work we extend the ideas of disjunctive programming, originally developed for mixed integer linear optimization, to the case of MISOCO problems. The extension presented here results in the derivation of a novel methodology that we call \emph{disjunctive conic cuts} for MISOCO problems. The analysis developed in this thesis is separated in three parts. In the first part, we introduce the formal definition of disjunctive conic cuts. Additionally, we show that under some mild assumptions there is a necessary and sufficient condition that helps to identify a disjunctive conic cut for a given convex set. The main appeal of this condition is that it can be easily verified in the case of MISOCO problems. In the second part, we study the geometry of sets defined by a single quadratic inequality. We show that for some of these sets it is possible to derive a close form to build a disjunctive conic cut. In the third part, we show that the feasible set of a MISOCO problem with a single cone can be characterized using sets that are defined by a single quadratic inequality. Then, we present the results that provide the criteria for the derivation of disjunctive conic cuts for MISOCO problems. Preliminary numerical experiments with our disjunctive conic cuts used in a branch-and-cut framework provide encouraging results where this novel methodology helped to solve MISOCO problems more efficiently. We close our discussion in this thesis providing some highlights about the questions that we consider worth pursuing for future research
Exploiting Structures in Mixed-Integer Second-Order Cone Optimization Problems for Branch-and-Conic-Cut Algorithms
This thesis studies computational approaches for mixed-integer second-order cone optimization (MISOCO) problems. MISOCO models appear in many real-world applications, so MISOCO has gained significant interest in recent years. However, despite recent advancements, there is a gap between the theoretical developments and computational practice. Three chapters of this thesis address three areas of computational methodology for an efficient branch-and-conic-cut (BCC) algorithm to solve MISOCO problems faster in practice. These chapters include a detailed discussion on practical work on adding cuts in a BCC algorithm, novel methodologies for warm-starting second-order cone optimization (SOCO) subproblems, and heuristics for MISOCO problems.The first part of this thesis concerns the development of a novel warm-starting method of interior-point methods (IPM) for SOCO problems. The method exploits the Jordan frames of an original instance and solves two auxiliary linear optimization problems. The solutions obtained from these problems are used to identify an ideal initial point of the IPM. Numerical results on public test sets indicate that the warm-start method works well in practice and reduces the number of iterations required to solve related SOCO problems by around 30-40%.The second part of this thesis presents novel heuristics for MISOCO problems. These heuristics use the Jordan frames from both continuous relaxations and penalty problems and present a way of finding feasible solutions for MISOCO problems. Numerical results on conic and quadratic test sets show significant performance in terms of finding a solution that has a small gap to optimality.The last part of this thesis presents application of disjunctive conic cuts (DCC) and disjunctive cylindrical cuts (DCyC) to asset allocation problems (AAP). To maximize the benefit from these powerful cuts, several decisions regarding the addition of these cuts are inspected in a practical setting. The analysis in this chapter gives insight about how these cuts can be added in case-specific settings
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