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SOLVING SPECIAL NONLINEAR FRACTIONAL-programming PROBLEMS VIA parametric LINEAR-programming

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1986年第3期23卷 396-400页

作者： WOLF, H FERNUNIV HAGEN FACHBEREICH WIRTSCH WISSENSCHD-5800 HAGENFED REP GER

A new procedure is developed to solve a generalized linear fractional programming problem. We find the optimal solutions in two steps. First we solve a parametric linear programming problem. Using the results of this step we then define a simple optimization problem in the second step. This yields an optimal value which together with the results of the parametric analysis provides the optimal solutions of the considered fractional programming problem. [ABSTRACT FROM AUTHOR]

关键词： fractional programming Optimization parametric programming

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Three kinds of convexity and concavity properties of optimal value function in parametric nonlinear programming

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OPTIMIZATION LETTERS 2012年第4期6卷 655-663页

作者： Wu Junliang Wang Yong Chongqing Univ Coll Math & Stat Chongqing 400044 Peoples R China Chongqing Univ Sch Econ & Business Adm Chongqing 400044 Peoples R China

This work is concerned with exploring the new convexity and concavity properties of the optimal value function in parametric programming. Some convex (concave) functions are discussed and sufficient conditions for new convexity and concavity of the optimal value function in parametric programming are given. Many results in this paper can be considered as deepen the convexity and concavity studies of convex (concave) functions and the optimal value functions.

关键词： parametric programming Optimal value function Convexity and concavity Set-valued map

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ALGORITHMS FOR parametric NONCONVEX programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1982年第3期38卷 319-340页

作者： BENSON, HP Associate Professor College of Business Administration University of Florida Gainesville Florida

In this paper, we consider a general family of nonconvex programming problems. All of the objective functions of the problems in this family are identical, but their feasibility regions depend upon a parameter ϑ. This family of problems is called a parametric nonconvex program (PNP). Solving (PNP) means finding an optimal solution for every program in the family. A prototype branch-and-bound algorithm is presented for solving (PNP). By modifying a prototype algorithm for solving a single nonconvex program, this algorithm solves (PNP) in one branch-and-bound search. To implement the algorithm, certain compact partitions and underestimating functions must be formed in an appropriate manner. We present an algorithm for solving a particular (PNP) which implements the prototype algorithm by forming compact partitions and underestimating functions based upon rules given by Falk and Soland. The programs in this (PNP) have the same concave objective function, but their feasibility regions are described by linear constraints with differing right-hand sides. Computational experience with this algorithm is reported for various problems.

关键词： Nonconvex programming parametric programming branch-and-bound methods global minimization

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Application of parametric programming based on interval number in risk management of commercial bank

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Dianzi Keji Daxue Xuebao/Journal of the University of Electronic Science and Technology of China 2006年第1期35卷 137-139页

作者： Guo, Zhan-Qin Zhou, Zong-Fang School of Management Science Univ. of Electron. Sci. and Technol. of China Chengdu 610054 China

A parametric programming based on interval number is proposed for the uncertainty risk and return. By selecting the risk parameters, commercial banks can get the optimum solution under the equilibrium of risk and return and can obtain the maximum return when risk is fixed. An example shows that the method is quite fit for credit risk management of commercial banks.

关键词： Credit risk management Interval number parametric programming Risk parameters

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A MULTI parametric QUADRATIC programming SOLUTION TO ROBUST PREDICTIVE CONTROL

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IFAC Proceedings Volumes 2005年第1期38卷 347-352页

作者： J.A. Rossiter B. Pluymers J.A.K. Suykens B. De Moor Research Group SCD-SISTA Department Of Electrotechnical Engineering (ESAT) Katholieke Universiteit Leuven Kasteelpark Arenberg 10 3001 Leuven Belgium Department of Automatic Control and Systems Engineering Mappin Street University of Sheffield S1 3JD UK

Multi parametric quadratic programming is an alternative means of implementing conventional predictive control algorithms whereby one transfers much of the computational load to offline calculations. This paper demonstrates how one can formulate a robust MPC problem as a quadratic program and hence make it amenable to MPQP solutions. The paper then derives some MPQP solutions and discusses the efficacy of these.

关键词： parametric programming quadratic programming robust MPC

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A Methodology for Building Generation Trajectories to Balance Continuous-Time Load Profiles

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IEEE TRANSACTIONS ON POWER SYSTEMS 2025年第2期40卷 1609-1622页

作者： Zhang, Menghan Zhou, Yue Xiang, Mingxu Yu, Juan Li, Wenyuan Yang, Zhifang Chongqing Univ Sch Elect Engn State Key Lab Power Transmiss Equipment Technol Chongqing 400044 Peoples R China Cardiff Univ Sch Engn Cardiff CF10 3AT Wales

In power systems, maintaining a balance between generation and load is crucial. Traditional discrete-time dispatch methods often fall short, as they do not account for continuous-time changes in the load profiles throughout the time span. This oversight can lead to inaccuracies in tracing load profiles and even cause ramping resource shortages. In this paper, we propose the idea of continuous-time generation trajectories as dispatch results, to align with continuous-time load profiles. To ensure the solvability of the continuous-time dispatch, we propose an iterative dispatch methodology, which includes two stages: trajectory construction and constraint verification. In the trajectory construction stage, we use a parametric programming model to divide the continuous-time load profiles into multiple segments. Subsequently, we build the generation trajectories for each segment using parametric solutions. In the constraint verification stage, we specifically check the continuous-time ramping constraints. This stage identifies the infeasible trajectories, which will be updated during the next iteration. We repeat this iterative process until each unit has a feasible continuous-time generation trajectory throughout the time span. The effectiveness of our methodology is demonstrated in an illustrative 5-bus system and an actual 661-bus system.

关键词： Trajectory Load modeling Uncertainty Real-time systems Predictive models Mathematical models Iterative methods Continuous-time dispatch continuous-time generation trajectories discrete-time dispatch parametric programming

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A Continuous Operating Envelope for Managing Intra-Interval Fluctuations: Modeling and Solution

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JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY 2025年第2期13卷 426-438页

作者： Zhang, Menghan Yang, Zhifang Yu, Juan Li, Wenyuan Chongqing Univ Sch Elect Engn State Key Lab Power Transmiss Equipment Technol Chongqing 400044 Peoples R China

Maintaining a continuous power balance is crucial for ensuring operational feasibility in power systems. However, due to forecasting difficulties and computational limitations, economic dispatch often relies on discrete interval horizons, which fail to guarantee feasibility within each interval. This paper introduces the concept of a continuous operating envelope for managing intra-interval fluctuations, delineating the range within which fluctuations remain manageable. We propose a parametric programming model to construct the envelope, represented as a polytope that accounts for both timescale and fluctuation dimensions. To address the computational challenges inherent in the parametric programming model, we develop a fast solution method to provide an approximated polytope. The approximated polytope, initially derived from lower-dimensional projections, represents a subset of the exact polytope that ensures operational feasibility. Additionally, we apply a polytope expansion strategy in the original dimensions to refine the approximated polytope, bringing the approximation closer to the exact polytope. Case studies on an illustrative 5-bus and a utility-scale 661-bus system demonstrate that the method effectively and stably provides a continuous operating envelope, particularly for high-dimensional problems.

关键词： Fluctuations Vectors Power systems programming Computational modeling Security Load flow Green energy Analytical models Uncertainty Continuous operating envelope economic dispatch intra-interval fluctuation feasibility uncertainty polytope parametric programming ancillary service

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A parametric APPROACH TO FUZZY LINEAR-programming

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FUZZY SETS AND SYSTEMS 1986年第1期20卷 17-30页

作者： CARLSSON, C KORHONEN, P HELSINKI SCH ECON SF-00100 HELSINKIFINLAND

In this paper we discuss how to deal with decision problems that are described with LP models and formulated with elements of imprecision and uncertainty. More precisely, we will study LP models in which the parameters are not fully known but only with some degree of precision. Even with incomplete information the model builder (or model user) is normally able to give a realistic interval for the parameters of an LP model. For the constraint vector this is combined with some wishes or some leeway on the constraints. Even with ambiquity in the objective function, there is normally some preference ordering to be found among alternative ways of action. We will demonstrate that these modelling complications can be handled with the help of some results developed in the theory of fuzzy sets. After an overview of some central contributions to fuzzy linear programming, we will develop an LP model in which the parameters are not fully known, only with some degree of precision, and show that the model can be parametrised in such a way that the optimal solution becomes a function of the degree of precision. The fuzzy LP model derived in this way appears to be fairly easy to handle computationally, which is demonstrated with a numerical example.

关键词： Fuzzy LP parametric programming

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NECESSARY CONDITIONS FOR UPPER SEMICONTINUITY IN parametric SEMI-INFINITE programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1986年第1期48卷 65-79页

作者： COLGEN, R 1.Fachbereich Mathematik Johann Wolfgang Goethe Universit?t Frankfurt am Main Federal Republic of Germany

We treat semi-infinite optimization problems: Minimizep(x) subject tox ∈ ? m , anda(t,x) ≦b(t) for allt ∈T, whereT is a σ-compact topological space, andp,a,b are suitable (?∞, ∞]-valued functions on R m , respec... 详细信息

We treat semi-infinite optimization problems: Minimizep(x) subject tox ∈ ?^m, anda(t,x) ≦b(t) for allt ∈T, whereT is a σ-compact topological space, andp,a,b are suitable (?∞, ∞]-valued functions on R^m, respectively. Linear, convex, and quasi-convex semi-infinite programming are included in this concept. The main results of this paper are on the necessity of the compactness of the set of feasible points for (a,b), and the set of ?-optimal solutions for (p,a,b) for the (Hausdorff) upper semicontinuity of the feasible set-mapping in (a,b), and the ?-optimal solution-mapping in (p,a,b), respectively (where the parameter sets are provided with a suitable topology). Some more special results complete the paper.

关键词： Nonlinear programming semi-infinite programming parametric programming set-valued mappings

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Optimal prioritized infeasibility handling in model predictive control: parametric preemptive multiobjective linear programming approach

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2001年第2期109卷 385-413页

作者： Vada, J Slupphaug, O Johansen, TA Energos Trondheim Norway ABB Ind Oslo Norway Norwegian Univ Sci & Technol Dept Engn Cybernet N-7034 Trondheim Norway

All practical implementations of model-based predictive control (MPC) require a means to recover from infeasibility. We propose a strategy designed for linear state-space MPC with prioritized constraints. It relaxes optimally an infeasible MPC optimization problem into a feasible one by solving a single-objective linear program (LP) online in addition to the standard online MPC optimization problem at each sample. By optimal, it is meant that the violation of a lower prioritized constraint cannot be made less without increasing the violation of a higher prioritized constraint. The problem of computing optimal constraint violations is naturally formulated as a parametric preemptive multiobjective LP. By extending well-known results from parametric LP, the preemptive multiobjective LP is reformulated into an equivalent standard single-objective LP. An efficient algorithm for offline design of this LP is given, and the algorithm is illustrated on an example.

关键词： parametric programming preemptive programming infeasibility linear model predictive control

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