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Numerical solution of a class of non-convex variational problems by SQP

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2002年第5-6期23卷 573-587页

作者： Mach, J Charles Univ Math Inst Prague 18675 8 Czech Republic

non-convex variational problems can be solved numerically by sequential quadratic programming (SQP), after relaxation by means of Young measures. The approximate solutions obtained by SQP converge weakly to the solution of the relaxed problem. Strong convergence is provided by compact imbedding of the space of feasible mappings due to assumptions on the growth of the involved integrand. An illustrative two-dimensional example with a known exact solution is included.

关键词： relaxation successive approximations mathematical programming non-linear programming

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Numerical Strategies for Mixed-Integer Optimization of Power-Split and Gear Selection in Hybrid Electric Vehicles

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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 2023年第3期24卷 3194-3210页

作者： Ganesan, Anand Gros, Sebastien Murgovski, Nikolce Volvo Car Corp Res & Dev S-40531 Gothenburg Sweden Chalmers Univ Technol Dept Elect Engn S-41296 Gothenburg Sweden Norwegian Univ Sci & Technol Dept Engn Cybernet N-7491 Trondheim Norway

This paper presents numerical strategies for a computationally efficient energy management system that co-optimizes the power split and gear selection of a hybrid electric vehicle (HEV). We formulate a mixed-integer optimal control problem (MIOCP) that is transcribed using multiple-shooting into a mixed-integer nonlinear program (MINLP) and then solved by nonlinear model predictive control. We present two different numerical strategies, a Selective Relaxation Approach (SRA), which decomposes the MINLP into several subproblems, and a Round-n-Search Approach (RSA), which is an enhancement of the known "relax-n-round' strategy. Subsequently, the resulting algorithmic performance and optimality of the solution of the proposed strategies are analyzed against two benchmark strate-gies;one using rule-based gear selection, which is typically used in production vehicles, and the other using dynamic programming (DP), which provides a global optimum of a quantized version of the MINLP. The results show that both SRA and RSA enable about 3.6% cost reduction compared to the rule-based strategy, while still being within 1% of the DP solution. Moreover, for the case studied RSA takes about 35% less mean computation time compared to SRA, while both SRA and RSA being about 99 times faster than DP. Furthermore, both SRA and RSA were able to overcome the infeasibilities encountered by a typical rounding strategy under different drive cycles. The results show the computational benefit of the proposed strategies, as well as the energy saving possibility of co-optimization strategies in which actuator dynamics are explicitly included.

关键词： Mixed-Integer nonlinear optimal control non-linear programming numerical optimization hybrid electric vehicle energy management nonlinear MPC optimal torque-split optimal gear selection

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Optimization methods for the verification of second order sufficient conditions for bang-bang controls

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OPTIMAL CONTROL APPLICATIONS & METHODS 2005年第3期26卷 129-156页

作者： Maurer, H Büskens, C Kim, JHR Kaya, CY Univ Munster Inst Numer Math D-48149 Munster Germany Univ Bremen Zentrum Technomath Fachbereich 3 Bremen Germany Univ S Australia Sch Math & Stat Mawson Lakes SA 5095 Australia

It has been common practice to find controls satisfying only necessary conditions for optimality, and then to use these controls assuming that they are (locally) optimal. However, sufficient conditions need to be used to ascertain that the control rule is optimal. Second order sufficient conditions (SSC) which have recently been derived by Agrachev, Stefani, and Zezza, and by Maurer and Osmolovskii, are a special form of sufficient conditions which are particularly suited for numerical verification. In this paper we present optimization methods and describe a numerical scheme for finding optimal bang-bang controls and verifying SSC. A straightforward transformation of the bang-bang arc durations allows one to use standard optimal control software to find the optimal are durations as well as to check SSC. The proposed computational verification technique is illustrated on three example applications. Copyright (c) 2005 John Wiley & Sons, Ltd.

关键词： bang-bang control second order sufficient conditions non-linear programming control of a van der Pol oscillator control of a batch reactor robot control

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Optimum design of plane timber trusses considering joint flexibility

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ENGINEERING STRUCTURES 2005年第1期27卷 145-154页

作者： Silih, S Premrov, M Kravanja, S Univ Maribor Fac Civil Engn SLO-2000 Maribor Slovenia

The paper presents the optimization of metal-plate-connected plane timber trusses with special emphasis on joint flexibility. The optimization was performed by the non-linear programming approach. Since various truss design parameters such as type of truss configuration, span/depth ratio, number and type of diagonal and vertical members and type of joint connections simultaneously affect each other, it is proposed that all of these parameters should be considered simultaneously in a single mathematical model. An optimization model for cost optimization of timber trusses was thus developed. The economic objective function for minimizing the structure's self-manufacturing costs was defined, subjected to the design, stress and deflection (in)equality constraints. The finite element equations were as the equality constraints defined for the calculation of the internal forces and the deflections of the structure. The stiffness matrix of the structure was composed by considering fictitiously decreased cross-section areas of all the flexibly connected elements. Constraints for the dimensioning of the timber members were determined in accordance with Eurocode 5 in order to satisfy the requirements of both the ultimate and the. serviceability limit states. The cross-section dimensions and the number of fasteners were defined as independent optimization variables. A numerical example demonstrates the applicability of the optimization approach presented as well as the influence of the fasteners' flexibility on the optimal self-manufacturing costs. (C) 2004 Elsevier Ltd. All rights reserved.

关键词： timber trusses fasteners flexibility optimization non-linear programming

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A non-linear possibilistic regression approach to model functional relationships in product planning

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2006年第11-12期28卷 1175-1181页

作者： Chen, YZ Chen, L Shanghai Univ Dept Precis Mech Engn Sch Mechatron Engn & Automat Shanghai 200444 Peoples R China

Product planning is one of four important processes in new product development (NPD) using quality function deployment (QFD). In order to model the process of product planning, the first problem to be solved is how to incorporate both qualitative and quantitative information regarding relationships between customer requirements and engineering characteristics, as well as among engineering characteristics, into the problem formulation. The inherent fuzziness of functional relationships in product planning makes the use of possibilistic regression justifiable. However, when linear programming in possibilistic regression analysis is used, some coefficients tend to become crisp because of the characteristic of linear programming. To tackle the problem, a non-linear programming based possibilistic regression approach is proposed, by which more diverse spread coefficients can be obtained than from a linear programming approach. An emulsification dynamite packing-machine design is used to illustrate the performance of the proposed approach.

关键词： house of quality non-linear programming possibilistic regression product planning quality function deployment triangular fuzzy number

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Solving knapsack problems with S-curve return functions

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2009年第2期193卷 605-615页

作者： Agrali, Semra Geunes, Joseph Univ Florida Dept Ind & Syst Engn Gainesville FL 32611 USA

We consider the allocation of a limited budget to a set of activities or investments in order to maximize return from investment. In a number of practical contexts (e.g., advertising), the return from investment in an activity is effectively modeled using an S-curve, where increasing returns to scale exist at small investment levels, and decreasing returns to scale occur at high investment levels. We demonstrate that the resulting knapsack problem with S-curve return functions is NP-hard, provide a pseudo-polynomial time algorithm for the integer variable version of the problem, and develop efficient solution methods for special cases of the problem. We also discuss a fully-polynomial-time approximation algorithm for the integer variable version of the problem. (C) 2007 Elsevier B.V. All rights reserved.

关键词： non-linear programming OR in strategic planning Dynamic programming Marketing

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A comparison of alternative models for solving a non-linear single plant Hydro Unit Commitment problem

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COMPUTERS & OPERATIONS RESEARCH 2024年 165卷

作者： Heintzmann, Alexandre Artigues, Christian Bendotti, Pascale Ngueveu, Sandra Ulrich Rottner, Cecile EDF Lab Paris Saclay 7 Bd Gaspard Monge F-91120 Palaiseau France Univ Toulouse LAAS CNRS CNRS INP Toulouse France

A wide range of real world optimization problems involves continuous decisions and non-linearities. Each nonlinear component of such problems can be modeled either linearly or non -linearly, considering or not additional integer variables. This results into different modeling choices that can drastically impact the solution time and quality. In this paper, we evaluate representative modeling alternatives, including common models from the literature as well as new models featuring less common functions. The single plant Hydro Unit Commitment problem (1-HUC) is the considered non-linear use case. Among the non-linearities of the 1-HUC, we focus on those involved in the power production, more precisely the head effect and the turbines efficiency. The power is defined as a two-dimensional non -convex and non -concave function of the water flow and head decision variables, the latter being itself a one-dimensional concave function of the turbined volume. We consider both the general problem and a common special case, assuming that the water head is fixed. Several available solvers are used for each non-linear model and the best virtual solver is retained to focus on the model capabilities rather than on the solver performance. Based on the numerical experiments, three models stand out as the most efficient in terms of computational time, solution quality and feasibility, sometimes in a counter -intuitive manner. For each of these models, a solver is highlighted as the most adequate.

关键词： non-linear programming non-linear modeling Hydro Unit Commitment

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Optimization of lift gas allocation in a gas lifted oil field as non-linear optimization problem

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MODELING IDENTIFICATION AND CONTROL 2012年第1期33卷 13-25页

作者： Sharma, Roshan Fjalestad, Kjetil Glemmestad, Bjorn Telemark Univ Coll Dept Elect Informat & Cubernet Porsgrunn Norway Statoil Res Ctr Porsgrunn Norway

Proper allocation and distribution of lift gas is necessary for maximizing total oil production from a field with gas lifted oil wells. When the supply of the lift gas is limited, the total available gas should be optimally distributed among the oil wells of the field such that the total production of oil from the field is maximized. This paper describes a non-linear optimization problem with constraints associated with the optimal distribution of the lift gas. A non-linear objective function is developed using a simple dynamic model of the oil field where the decision variables represent the lift gas flow rate set points of each oil well of the field. The lift gas optimization problem is solved using the 'fmincon' solver found in MATLAB. As an alternative and for verification, hill climbing method is utilized for solving the optimization problem. Using both of these methods, it has been shown that after optimization, the total oil production is increased by about 4%. For multiple oil wells sharing lift gas from a common source, a cascade control strategy along with a nonlinear steady state optimizer behaves as a self-optimizing control structure when the total supply of lift gas is assumed to be the only input disturbance present in the process. Simulation results show that repeated optimization performed after the first time optimization under the presence of the input disturbance has no effect in the total oil production.

关键词： Optimization non-linear programming cascade control structure gas lifted oil well hill climbing self-optimization

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RECYCLING OF LIFETIME DEPENDENT DETERIORATED PRODUCTS THROUGH DIFFERENT SUPPLY CHAINS

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RAIRO-OPERATIONS RESEARCH 2019年第1期53卷 129-156页

作者： Iqbal, Muhammad Waqas Sarkar, Biswajit Hanyang Univ Dept Ind & Management Engn Ansan 15588 Gyeonggi Do South Korea

Conservation of natural resources in order to protect the environment, support the economy, and offer a better life to living beings has become an urgent need of modern business so that future generations can survive within available resources and in a healthy environment. Recycling of used products plays an important role in the conservation of natural resources and the development of sustainable business for deteriorating products because the number of these items increases with time, which creates economic loss and environmental pollution. This paper considers the production and cycle time as decision variables to design a forward and reverse supply chain system that produces two different types of products, which are subject to deterioration. Rate of deterioration is time-varying and depends on the maximum lifetime of products. Used products of a forward supply chain are collected and treated as raw materials in a reverse supply chain to produce other products. The system involves three types of inventory stocks, i.e., product 1, 2, and returned inventory. The objective of this research is to minimize total cost per unit time for two types of systems, one in which products of both the supply chains deteriorate and the second in which the products of the first supply chain deteriorate. Kuhn-Tucker method is employed to solve the model and a solution algorithm is proposed to obtain optimal solution. Application of the model is supported with numerical examples and sensitivity analysis. Some managerial insights are provided to help managers while applying the proposed models in real situations. Results of numerical experiments suggest for deteriorating products to plan short replenishment cycles of inventory.

关键词： Supply chain management time-varying deterioration reused products maximum lifetime demand dependent production rate non-linear programming

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To be fair or efficient or a bit of both

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COMPUTERS & OPERATIONS RESEARCH 2008年第12期35卷 3787-3806页

作者： Zukerman, Moshe Mammadov, Musa Tan, Liansheng Ouveysi, Iradj Andrew, Lachlan L. H. Univ Melbourne Dept Elect & Elect Engn ARC Special Res Ctr Ultra Broadband Informat Netw Melbourne Vic 3010 Australia Univ Ballarat Sch Informat Technol & Math Sci Ballarat Vic 3353 Australia Cent China Normal Univ Dept Comp Sci Wuhan 430079 Peoples R China CALTECH Dept Comp Sci Pasadena CA 91125 USA

Introducing a new concept of (alpha, beta)-fairness, which allows for a bounded fairness compromise, so that a source is allocated a rate neither less than 0 = 1, times its fair share, this paper provides a framework ... 详细信息

Introducing a new concept of (alpha, beta)-fairness, which allows for a bounded fairness compromise, so that a source is allocated a rate neither less than 0 <=alpha <= 1, not more than beta >= 1, times its fair share, this paper provides a framework to optimize efficiency (utilization, throughput or revenue) subject to fairness constraints in a general telecommunications network for an arbitrary fairness criterion and cost functions. We formulate a non-linear program (NLP) that finds the optimal bandwidth allocation by maximizing efficiency subject to (alpha, beta)-fairness constraints. This leads to what we call an efficiency-faimess function, which shows the benefit in efficiency as a function of the extent to which fairness is compromised. To solve the NLP we use two algorithms. The first is a well-known branch-and-bound-based algorithm called Lipschitz Global Optimization and the second is a recently developed algorithm called Algorithm for Global Optimization Problems (AGOP). We demonstrate the applicability of the framework to a range of examples from sharing a single link to efficiency fairness issues associated with serving customers in remote communities.(c) 2007 Elsevier Ltd. All rights reserved.

关键词： non-linear programming utility optimization fairness efficiency-fairness tradeoff bandwidth allocation

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