3,438 research outputs found
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Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation
A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined
Extended Formulations in Mixed-integer Convex Programming
We present a unifying framework for generating extended formulations for the
polyhedral outer approximations used in algorithms for mixed-integer convex
programming (MICP). Extended formulations lead to fewer iterations of outer
approximation algorithms and generally faster solution times. First, we observe
that all MICP instances from the MINLPLIB2 benchmark library are conic
representable with standard symmetric and nonsymmetric cones. Conic
reformulations are shown to be effective extended formulations themselves
because they encode separability structure. For mixed-integer
conic-representable problems, we provide the first outer approximation
algorithm with finite-time convergence guarantees, opening a path for the use
of conic solvers for continuous relaxations. We then connect the popular
modeling framework of disciplined convex programming (DCP) to the existence of
extended formulations independent of conic representability. We present
evidence that our approach can yield significant gains in practice, with the
solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201
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Linear, integer separable and fuzzy programming problems: a united approach towards automatic reformulation
For mathematical programming (MP) to have greater impact as a
decision tool, MP software systems must offer suitable support in
terms of model communication and modelling techniques. In this
paper modelling techniques that allow logical restrictions to be
modelled in integer programming terms are described and their
implications discussed. In addition it is demonstrated that many
classes of non-linearities which are not variable separable may be
after suitable algebraic manipulation put in a variable separable
form. The methods of reformulating the fuzzy linear programming
problem as a Max-Min problem is also introduced. It is shown that
analysis of bounds plays a key role in the following four important
contexts: model reduction, reformulation of logical restrictions
as 0-1 mixed integer programs, reformulation of nonlinear programs
as variable separable programs and reformulation of fuzzy linear
programs. It is observed that as well as incorporating an
interface between the modeller and the optimiser there is a need to
make available to the modeller software facilities which support the
model reformulation techniques described here
Decomposition, Reformulation, and Diving in University Course Timetabling
In many real-life optimisation problems, there are multiple interacting
components in a solution. For example, different components might specify
assignments to different kinds of resource. Often, each component is associated
with different sets of soft constraints, and so with different measures of soft
constraint violation. The goal is then to minimise a linear combination of such
measures. This paper studies an approach to such problems, which can be thought
of as multiphase exploitation of multiple objective-/value-restricted
submodels. In this approach, only one computationally difficult component of a
problem and the associated subset of objectives is considered at first. This
produces partial solutions, which define interesting neighbourhoods in the
search space of the complete problem. Often, it is possible to pick the initial
component so that variable aggregation can be performed at the first stage, and
the neighbourhoods to be explored next are guaranteed to contain feasible
solutions. Using integer programming, it is then easy to implement heuristics
producing solutions with bounds on their quality.
Our study is performed on a university course timetabling problem used in the
2007 International Timetabling Competition, also known as the Udine Course
Timetabling Problem. In the proposed heuristic, an objective-restricted
neighbourhood generator produces assignments of periods to events, with
decreasing numbers of violations of two period-related soft constraints. Those
are relaxed into assignments of events to days, which define neighbourhoods
that are easier to search with respect to all four soft constraints. Integer
programming formulations for all subproblems are given and evaluated using ILOG
CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table
Using Functional Programming to recognize Named Structure in an Optimization Problem: Application to Pooling
Branch-and-cut optimization solvers typically apply generic algorithms, e.g., cutting planes or primal heuristics, to expedite performance for many mathematical optimization problems. But solver software receives an input optimization problem as vectors of equations and constraints containing no structural information. This article proposes automatically detecting named special structure using the pattern matching features of functional programming. Specifically, we deduce the industrially-relevant nonconvex nonlinear Pooling Problem within a mixed-integer nonlinear optimization problem and show that we can uncover pooling structure in optimization problems which are not pooling problems. Previous work has shown that preprocessing heuristics can find network structures; we show that we can additionally detect nonlinear pooling patterns. Finding named structures allows us to apply, to generic optimization problems, cutting planes or primal heuristics developed for the named structure. To demonstrate the recognition algorithm, we use the recognized structure to apply primal heuristics to a test set of standard pooling problems
Robust Model Predictive Control for Signal Temporal Logic Synthesis
Most automated systems operate in uncertain or adversarial conditions, and have to be capable of reliably reacting to changes in the environment. The focus of this paper is on automatically synthesizing reactive controllers for cyber-physical systems subject to signal temporal logic (STL) specifications. We build on recent work that encodes STL specifications as mixed integer linear constraints on the variables of a discrete-time model of the system and environment dynamics. To obtain a reactive controller, we present solutions to the worst-case model predictive control (MPC) problem using a suite of mixed integer linear programming techniques. We demonstrate the comparative effectiveness of several existing worst-case MPC techniques, when applied to the problem of control subject to temporal logic specifications; our empirical results emphasize the need to develop specialized solutions for this domain
Evaluating Resilience of Electricity Distribution Networks via A Modification of Generalized Benders Decomposition Method
This paper presents a computational approach to evaluate the resilience of
electricity Distribution Networks (DNs) to cyber-physical failures. In our
model, we consider an attacker who targets multiple DN components to maximize
the loss of the DN operator. We consider two types of operator response: (i)
Coordinated emergency response; (ii) Uncoordinated autonomous disconnects,
which may lead to cascading failures. To evaluate resilience under response
(i), we solve a Bilevel Mixed-Integer Second-Order Cone Program which is
computationally challenging due to mixed-integer variables in the inner problem
and non-convex constraints. Our solution approach is based on the Generalized
Benders Decomposition method, which achieves a reasonable tradeoff between
computational time and solution accuracy. Our approach involves modifying the
Benders cut based on structural insights on power flow over radial DNs. We
evaluate DN resilience under response (ii) by sequentially computing autonomous
component disconnects due to operating bound violations resulting from the
initial attack and the potential cascading failures. Our approach helps
estimate the gain in resilience under response (i), relative to (ii)
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Tools for reformulating logical forms into zero-one mixed integer programs (MIPS)
A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Programming (ILP) formulation or a Mixed Integer Programming (MIP) formulation is presented. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. A prototype user interface by which logical forms can be reformulated and the corresponding MIP constructed and analysed within an existing Mathematical Programming modelling system is illustrated. Finally, the steps to formulate a discrete optimisation model in this way are demonstrated by means of an example
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