599 research outputs found
Global optimisation of large-scale quadratic programs: application to short-term planning of industrial refinery-petrochemical complexes
This thesis is driven by an industrial problem arising in the short-term planning of an integrated refinery-petrochemical complex (IRPC) in Colombia. The IRPC of interest is composed of 60 industrial plants and a tank farm for crude mixing and fuel blending consisting of 30 additional units. It considers both domestic and imported crude oil supply, as well as refined product imports such as low sulphur diesel and alkylate. This gives rise to a large-scale mixed-integer quadratically constrained quadratic program (MIQCQP) comprising about 7,000 equality constraints with over 35,000 bilinear terms and 280 binary variables describing operating modes for the process units. Four realistic planning scenarios are recreated to study the performance of the algorithms developed through the thesis and compare them to commercial solvers.
Local solvers such as SBB and DICOPT cannot reliably solve such large-scale MIQCQPs. Usually, it is challenging to even reach a feasible solution with these solvers, and a heuristic procedure is required to initialize the search. On the other hand, global solvers such as ANTIGONE and BARON determine a feasible solution for all the scenarios analysed, but they are unable to close the relaxation gap to less than 40% on average after 10h of CPU runtime. Overall, this industrial-size problem is thus intractable to global optimality in a monolithic way.
The first main contribution of the thesis is a deterministic global optimisation algorithm based on cluster decomposition (CL) that divides the network into groups of process units according to their functionality. The algorithm runs through the sequences of clusters and proceeds by alternating between: (i) the (global) solution of a mixed-integer linear program (MILP), obtained by relaxing the bilinear terms based on their piecewise McCormick envelopes and a dynamic partition of their variable ranges, in order to determine an upper bound on the maximal profit; and (ii) the local solution of a quadratically-constrained quadratic program (QCQP), after fixing the binary variables and initializing the continuous variables to the relaxed MILP solution point, in order to determine a feasible solution (lower bound on the maximal profit). Applied to the base case scenario, the CL approach reaches a best solution of 2.964 MMUSD/day and a relaxation gap of 7.5%, a remarkable result for such challenging MIQCQP problem. The CL approach also vastly outperforms both ANTIGONE (2.634 MMUSD/day, 32% optimality gap) and BARON (2.687 MMUSD/day, 40% optimality gap).
The second main contribution is a spatial Lagrangean decomposition, which entails decomposing the IRPC short-term planning problem into a collection of smaller subproblems that can be solved independently to determine an upper bound on the maximal profit. One advantage of this strategy is that each sub-problem can be solved to global optimality, potentially providing good initial points for the monolithic problem itself. It furthermore creates a virtual market for trading crude blends and intermediate refined–petrochemical streams and seeks an optimal trade-off in such a market, with the Lagrange multipliers acting as transfer prices. A decomposition over two to four is considered, which matches the crude management, refinery, petrochemical operations, and fuel blending sections of the IRPC. An optimality gap below 4% is achieved in all four scenarios considered, which is a significant improvement over the cluster decomposition algorithm.Open Acces
Bilevel optimisation with embedded neural networks: Application to scheduling and control integration
Scheduling problems requires to explicitly account for control considerations
in their optimisation. The literature proposes two traditional ways to solve
this integrated problem: hierarchical and monolithic. The monolithic approach
ignores the control level's objective and incorporates it as a constraint into
the upper level at the cost of suboptimality. The hierarchical approach
requires solving a mathematically complex bilevel problem with the scheduling
acting as the leader and control as the follower. The linking variables between
both levels belong to a small subset of scheduling and control decision
variables. For this subset of variables, data-driven surrogate models have been
used to learn follower responses to different leader decisions. In this work,
we propose to use ReLU neural networks for the control level. Consequently, the
bilevel problem is collapsed into a single-level MILP that is still able to
account for the control level's objective. This single-level MILP reformulation
is compared with the monolithic approach and benchmarked against embedding a
nonlinear expression of the neural networks into the optimisation. Moreover, a
neural network is used to predict control level feasibility. The case studies
involve batch reactor and sequential batch process scheduling problems. The
proposed methodology finds optimal solutions while largely outperforming both
approaches in terms of computational time. Additionally, due to well-developed
MILP solvers, adding ReLU neural networks in a MILP form marginally impacts the
computational time. The solution's error due to prediction accuracy is
correlated with the neural network training error. Overall, we expose how - by
using an existing big-M reformulation and being careful about integrating
machine learning and optimisation pipelines - we can more efficiently solve the
bilevel scheduling-control problem with high accuracy.Comment: 18 page
Alive Caricature from 2D to 3D
Caricature is an art form that expresses subjects in abstract, simple and
exaggerated view. While many caricatures are 2D images, this paper presents an
algorithm for creating expressive 3D caricatures from 2D caricature images with
a minimum of user interaction. The key idea of our approach is to introduce an
intrinsic deformation representation that has a capacity of extrapolation
enabling us to create a deformation space from standard face dataset, which
maintains face constraints and meanwhile is sufficiently large for producing
exaggerated face models. Built upon the proposed deformation representation, an
optimization model is formulated to find the 3D caricature that captures the
style of the 2D caricature image automatically. The experiments show that our
approach has better capability in expressing caricatures than those fitting
approaches directly using classical parametric face models such as 3DMM and
FaceWareHouse. Moreover, our approach is based on standard face datasets and
avoids constructing complicated 3D caricature training set, which provides
great flexibility in real applications.Comment: Accepted to CVPR 201
The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries
We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body
An Example of Clifford Algebras Calculations with GiNaC
This example of Clifford algebras calculations uses GiNaC
(http://www.ginac.de/) library, which includes a support for generic Clifford
algebra starting from version~1.3.0. Both symbolic and numeric calculation are
possible and can be blended with other functions of GiNaC. This calculations
was made for the paper math.CV/0410399.
Described features of GiNaC are already available at PyGiNaC
(http://sourceforge.net/projects/pyginac/) and due to course should propagate
into other software like GNU Octave (http://www.octave.org/), gTybalt
(http://www.fis.unipr.it/~stefanw/gtybalt.html), which use GiNaC library as
their back-end.Comment: 20 pages, LaTeX2e, 12 PS graphics in one figure; v3 code
improvements; v4 small code correction for new libraries; v5 comments are
redesined to be more readabl
Low-rank multi-parametric covariance identification
We propose a differential geometric construction for families of low-rank
covariance matrices, via interpolation on low-rank matrix manifolds. In
contrast with standard parametric covariance classes, these families offer
significant flexibility for problem-specific tailoring via the choice of
"anchor" matrices for the interpolation. Moreover, their low-rank facilitates
computational tractability in high dimensions and with limited data. We employ
these covariance families for both interpolation and identification, where the
latter problem comprises selecting the most representative member of the
covariance family given a data set. In this setting, standard procedures such
as maximum likelihood estimation are nontrivial because the covariance family
is rank-deficient; we resolve this issue by casting the identification problem
as distance minimization. We demonstrate the power of these differential
geometric families for interpolation and identification in a practical
application: wind field covariance approximation for unmanned aerial vehicle
navigation
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