scholarly journals Computational Experience with Piecewise Linear Relaxations for Petroleum Refinery Planning

Processes ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1624
Author(s):  
Zaid Ashraf Rana ◽  
Cheng Seong Khor ◽  
Haslinda Zabiri
Keyword(s):  
Piecewise Linear ◽  
Mixed Integer ◽  
Integer Optimization ◽  
Solution Quality ◽  
Uniform Partition ◽  
Planning Optimization ◽  
Relaxation Schemes ◽  
Relaxation Solution ◽  
Refinery Planning ◽  
Crude Oil Distillation

Refinery planning optimization is a challenging problem as regards handling the nonconvex bilinearity, mainly due to pooling operations in processes such as crude oil distillation and product blending. This work investigated the performance of several representative piecewise linear (or piecewise affine) relaxation schemes (referred to as McCormick, bm, nf5, and nf6t) and de (which is a new approach proposed based on eigenvector decomposition) that mainly give rise to mixed-integer optimization programs to convexify a bilinear term using predetermined univariate partitioning for instances of uniform and non-uniform partition sizes. The computational results showed that applying these schemes improves the relaxation tightness compared to only applying convex and concave envelopes as estimators. Uniform partition sizes typically perform better in terms of relaxation solution quality and convergence behavior. It was also seen that there is a limit on the number of partitions that contribute to relaxation tightness, which does not necessarily correspond to a larger number of partitions, while a direct relationship between relaxation size and tightness does not always hold for non-uniform partition sizes.

scholarly journals Computational Experience with Piecewise-Linear Relaxations for Petroleum Refinery Planning

2021 ◽  
Author(s):  
Zaid Ashraf Rana ◽  
Cheng Seong Khor
Keyword(s):  
Piecewise Linear ◽  
Mixed Integer ◽  
Integer Optimization ◽  
Solution Quality ◽  
Uniform Partition ◽  
Planning Optimization ◽  
Relaxation Schemes ◽  
Relaxation Solution ◽  
Refinery Planning ◽  
Crude Oil Distillation

Refinery planning optimization is a challenging problem as regards handling the nonconvex bilinearity mainly due to pooling operations in processes such as crude oil distillation and product blending. This work investigates the performance of several representative piecewise-linear (or piecewise-affine) relaxation schemes (referred to as McCormick, bm, nf5, nf6t, and de (which is a new approach proposed based on eigenvector decomposition) that mainly give rise to mixed-integer optimization programs to convexify a bilinear term using predetermined univariate partitioning for instances of uniform and non-uniform partition sizes. Computational results show that applying these schemes give improved relaxation tightness than only applying convex and concave envelopes as estimators. Uniform partition sizes typically perform better in terms of relaxation solution quality and convergence behavior. It is also seen that there is a limit on the number of partitions that contributes to relaxation tightness, which does not necessarily correspond to a larger number of partitions, while a direct relation between relaxation size and tightness does not always hold for non-uniform partition sizes.

scholarly journals Sparse Poisson regression via mixed-integer optimization

PLoS ONE ◽  
2021 ◽  
Vol 16 (4) ◽  
pp. e0249916
Author(s):  
Hiroki Saishu ◽  
Kota Kudo ◽  
Yuichi Takano
Keyword(s):  
Poisson Regression ◽  
Likelihood Function ◽  
Piecewise Linear ◽  
Low Noise ◽  
Computer Hardware ◽  
Mixed Integer ◽  
Piecewise Linear Approximation ◽  
Integer Optimization ◽  
Mixed Integer Optimization ◽  
Log Likelihood

We present a mixed-integer optimization (MIO) approach to sparse Poisson regression. The MIO approach to sparse linear regression was first proposed in the 1970s, but has recently received renewed attention due to advances in optimization algorithms and computer hardware. In contrast to many sparse estimation algorithms, the MIO approach has the advantage of finding the best subset of explanatory variables with respect to various criterion functions. In this paper, we focus on a sparse Poisson regression that maximizes the weighted sum of the log-likelihood function and the L2-regularization term. For this problem, we derive a mixed-integer quadratic optimization (MIQO) formulation by applying a piecewise-linear approximation to the log-likelihood function. Optimization software can solve this MIQO problem to optimality. Moreover, we propose two methods for selecting a limited number of tangent lines effective for piecewise-linear approximations. We assess the efficacy of our method through computational experiments using synthetic and real-world datasets. Our methods provide better log-likelihood values than do conventional greedy algorithms in selecting tangent lines. In addition, our MIQO formulation delivers better out-of-sample prediction performance than do forward stepwise selection and L1-regularized estimation, especially in low-noise situations.

Nonlinear and Mixed-Integer Optimization

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Nonlinear Optimization ◽  
Systems Engineering ◽  
Chemical Engineering ◽  
Applied Mathematics ◽  
Linear Optimization ◽  
Mixed Integer ◽  
Process Synthesis ◽  
Synthesis Process ◽  
Integer Optimization ◽  
Mixed Integer Optimization

Filling a void in chemical engineering and optimization literature, this book presents the theory and methods for nonlinear and mixed-integer optimization, and their applications in the important area of process synthesis. Other topics include modeling issues in process synthesis, and optimization-based approaches in the synthesis of heat recovery systems, distillation-based systems, and reactor-based systems. The basics of convex analysis and nonlinear optimization are also covered and the elementary concepts of mixed-integer linear optimization are introduced. All chapters have several illustrations and geometrical interpretations of the material as well as suggested problems. Nonlinear and Mixed-Integer Optimization will prove to be an invaluable source--either as a textbook or a reference--for researchers and graduate students interested in continuous and discrete nonlinear optimization issues in engineering design, process synthesis, process operations, applied mathematics, operations research, industrial management, and systems engineering.

scholarly journals MINLP formulations for continuous piecewise linear function fitting

2021 ◽  
Author(s):  
Noam Goldberg ◽  
Steffen Rebennack ◽  
Youngdae Kim ◽  
Vitaliy Krasko ◽  
Sven Leyffer
Keyword(s):  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Function Fitting ◽  
Computational Performance ◽  
Integer Nonlinear Programming ◽  
Minlp Model ◽  
Necessary Constraints ◽  
Continuous Piecewise Linear Function

AbstractWe consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. 10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

A mixed-integer optimization approach for homogeneous magnet design

TECHNOLOGY ◽  
2018 ◽  
Vol 06 (02) ◽  
pp. 49-58
Author(s):  
Iman Dayarian ◽  
Timothy C.Y. Chan ◽  
David Jaffray ◽  
Teo Stanescu
Keyword(s):  
Imaging Modality ◽  
Spatial Location ◽  
Mixed Integer ◽  
Optimization Approach ◽  
Integer Optimization ◽  
Magnet Design ◽  
Field Homogeneity ◽  
Feasibility Constraints ◽  
Design Generation ◽  
Magnetic Resonance Imaging Mri

Magnetic resonance imaging (MRI) is a powerful diagnostic tool that has become the imaging modality of choice for soft-tissue visualization in radiation therapy. Emerging technologies aim to integrate MRI with a medical linear accelerator to form novel cancer therapy systems (MR-linac), but the design of these systems to date relies on heuristic procedures. This paper develops an exact, optimization-based approach for magnet design that 1) incorporates the most accurate physics calculations to date, 2) determines precisely the relative spatial location, size, and current magnitude of the magnetic coils, 3) guarantees field homogeneity inside the imaging volume, 4) produces configurations that satisfy, for the first time, small-footprint feasibility constraints required for MR-linacs. Our approach leverages modern mixed-integer programming (MIP), enabling significant flexibility in magnet design generation, e.g., controlling the number of coils and enforcing symmetry between magnet poles. Our numerical results demonstrate the superiority of our method versus current mainstream methods.

Split cuts for robust mixed-integer optimization

2012 ◽  
Vol 40 (3) ◽  
pp. 165-171
Author(s):  
Utz-Uwe Haus ◽  
Frank Pfeuffer
Keyword(s):  
Mixed Integer ◽  
Integer Optimization ◽  
Mixed Integer Optimization ◽  
Split Cuts
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