Many practical engineering problems, for example, in the areas of power systems, transportation, and data science, have physical aspects that are naturally modeled by smooth nonlinear functions, as well as design aspe...
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Many practical engineering problems, for example, in the areas of power systems, transportation, and data science, have physical aspects that are naturally modeled by smooth nonlinear functions, as well as design aspects that are often modeled via discrete decision variables. By combining ideas from continuous and discrete optimization, research in mixed-integer nonlinear optimization (MINLO) seeks to develop efficient and scalable exact or approximate algorithm frameworks attacking these challenging models. The dissertation focuses on treating some difficulties related to algorithm design and modeling principle in MINLO. First, in the context of the spatial branch-and-bound approach for factorable MINLO models, which decomposes complicated functions into compositions of affine functions and low-dimensional "library'' functions, we investigate a smoothing technique for univariate library functions that are not sufficiently smooth (i.e., not twice continuously differentiable). Next, we study a more tractable relaxation of a special MINLO model involving indicator variables and univariate convex functions, and we use volume as a measure for comparing the tightness of relaxations to balance the trade-off between tightness and tractability. We also lift some univariate results to the multivariate case on a simplex. Finally, we introduce various sparse generalized inverses and present an efficient and practical local-search algorithms to construct approximate sparse generalized inverses with both a low-rank property and a block structure.
We consider mixed-integernonlinear robust optimization problems with nonconvexities. In detail, the functions can be nonsmooth and generalized convex, i.e., f degrees-quasiconvex or f degrees-pseudoconvex. We propose...
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We consider mixed-integernonlinear robust optimization problems with nonconvexities. In detail, the functions can be nonsmooth and generalized convex, i.e., f degrees-quasiconvex or f degrees-pseudoconvex. We propose a robust optimization method that requires no certain structure of the adversarial problem, but only approximate worst-case evaluations. The method integrates a bundle method, for continuous subproblems, into an outer approximation approach. We prove that our algorithm converges and finds an approximately robust optimal solution and propose robust gas transport as a suitable application.
We introduce two new optimization models for the aircraft conflict avoidance problem that aims at issuing decisions on both speed and heading-angle deviations to keep aircraft pairwise separated by a given separation ...
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We introduce two new optimization models for the aircraft conflict avoidance problem that aims at issuing decisions on both speed and heading-angle deviations to keep aircraft pairwise separated by a given separation distance. The first model is a new mixed-integernonlinear formulation. The second model is a continuous optimization formulation, less typical in aircraft conflict avoidance. The advantages of the two models are combined within a three-phase method that we propose to solve the problem to global optimality. Computational experiments on various instances from the literature yield very promising results, and show the effectiveness of the proposed models and of the three-phase solution approach. (c) 2023 Published by Elsevier B.V.
Currently, few approaches are available for mixed-integernonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protect...
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Currently, few approaches are available for mixed-integernonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integernonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem. In our approach, we model a robust optimization problem as a nonsmooth mixed-integernonlinear problem and tackle it by an outer approximation method that requires only inexact function values and subgradients. To deal with the arising nonlinear subproblems, we render an adaptive bundle method applicable to this setting and extend it to generate cutting planes, which are valid up to a known precision. Relying on its convergence to approximate critical points, we prove, as a consequence, finite convergence of the outer approximation algorithm. As an application, we study the gas transport problem under uncertainties in demand and physical parameters on realistic instances and provide computational results demonstrating the efficiency of our method.
We present several mathematical-optimization formulations for a problem that commonly occurs in geometry processing and specifically in the design of so-called smooth direction fields on surfaces. This problem has dir...
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We present several mathematical-optimization formulations for a problem that commonly occurs in geometry processing and specifically in the design of so-called smooth direction fields on surfaces. This problem has direct applications in 3D shape parameterization, texture mapping, and shape design via rough concept sketches, among many others. A key challenge in this setting is to design a set of unit-norm directions, on a given surface, that satisfy some prescribed constraints and vary smoothly. This naturally leads to mixed-integeroptimization formulations, because the smoothness needs to be formulated with respect to angle-valued variables, which to compare one needs to fix the discrete jump between nearby points. Previous works have primarily attacked this problem via a greedy ad-hoc strategy with a specialized solver. We demonstrate how the problem can be cast in a standard mathematical-optimization form, and we suggest several relaxations that are especially adapted to modern mathematical-optimization solvers. .
Hard problems of discrete geometry may be formulated as a global optimization problems, which may be solved by general purpose solvers implementing branch-and-bound (B&B) algorithm. A problem of densest packing of...
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ISBN:
(纸本)9783030365929;9783030365912
Hard problems of discrete geometry may be formulated as a global optimization problems, which may be solved by general purpose solvers implementing branch-and-bound (B&B) algorithm. A problem of densest packing of N equal circles in special geometrical object, so called Square Flat Torus, R-2/Z(2), with the induced metric, is considered. It is formulated as mixed-integer problem with linear and nonconvex quadratic constraints. The open-source B&B-solver SCIP and its parallel implementation ParaSCIP have been used to find optimal arrangements for N <= 9. The main result is a confirmation of the conjecture on optimal packing for N = 9 that was published in 2012 by O. Musin and A. Nikitenko.
In this paper we propose to utilize a variation of the alternating direction method of multipliers (ADMM) as a simple heuristic for mixed-integer nonlinear optimization problems in structural optimization. Numerical e...
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In this paper we propose to utilize a variation of the alternating direction method of multipliers (ADMM) as a simple heuristic for mixed-integer nonlinear optimization problems in structural optimization. Numerical experiments suggest that using multiple restarts of ADMM with random initial points often yields a reasonable solution with small computational cost.
In this contribution a time constrained mixedintegernonlinearoptimization is presented aiming at optimal operation of coupled power and gas distribution grids. Optimal scheduling of Power-to-Gas and Combined Heat a...
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ISBN:
(纸本)9781538629109
In this contribution a time constrained mixedintegernonlinearoptimization is presented aiming at optimal operation of coupled power and gas distribution grids. Optimal scheduling of Power-to-Gas and Combined Heat and Power Systems enables a flexible operation of the resulting multi-carrier energy system. Linear constraints consider static models of the conversion technologies. The distribution grid models of power and gas system are included as nonlinear constraints. In a case study the resulting optimization framework is applied to a small test system based on real data.
This paper improves the adaptive metamodel-based global algorithm (AMGO), which is presented for unconstrained continuous problems, to solve mixed-integer nonlinear optimization involving black-box and expensive funct...
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This paper improves the adaptive metamodel-based global algorithm (AMGO), which is presented for unconstrained continuous problems, to solve mixed-integer nonlinear optimization involving black-box and expensive functions. The new proposed method is called as METADIR, which can be divided into two stages. In the first stage, the METADIR adopts extended DIRECT method to constantly subdivide the design space and identify the sub-region that may contain the optimal value. When iterative points gather into a sub-region to some extent, we terminate the search progress of DIRECT and turn to the next stage. In the second phase, a local metamodel is constructed in this potential optimal sub-region, and then an auxiliary optimization problem extended from AMGO is established based on the local metamodel to obtain the iterative points, which are then applied to update the metamodel adaptively. To show the performance of METADIR on both continuous and mixed-integer problems, numerical tests are presented on both kinds of problems. The METADIR method is compared with the original DIRECT on continuous problems, and compared with SO-MI and GA on mixed-integer problems. Test results show that the proposed method has better accuracy and needs less function evaluations. Finally, the new proposed method is applied into the component size optimization problem of fuel cell vehicle and achieves satisfied results.
We consider the design and operation of water networks simultaneously. Water network problems can be divided into two categories: the design problem and the operation problem. The design problem involves determining t...
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We consider the design and operation of water networks simultaneously. Water network problems can be divided into two categories: the design problem and the operation problem. The design problem involves determining the appropriate pipe sizing and placements of pump stations, while the operation problem involves scheduling pump stations over multiple time periods to account for changes in supply and demand. Our focus is on networks that involve water co-produced with oil and gas. While solving the optimization formulation for such networks, we found that obtaining a primal (feasible) solution is more challenging than obtaining dual bounds using off-the-shelf mixed-integernonlinear programming solvers. Therefore, we propose two methods to obtain good primal solutions. One method involves a decomposition framework that utilizes a convex reformulation, while the other is based on time decomposition. To test our proposed methods, we conduct computational experiments on a network derived from the PARETO case study.
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