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Convex underestimation strategies for signomial functions

OPTIMIZATION METHODS & SOFTWARE 2009年第4-5期24卷 505-522页

作者： Lundell, Andreas Westerlund, Tapio Abo Akad Univ Turku Finland

Different types of underestimation strategies are used in deterministic global optimization. In this paper, convexification and underestimation techniques applicable to problems containing signomial functions are studied. Especially, power transformation and exponential transformation (ET) will be considered in greater detail and some new theoretical results regarding the relation between the negative power transformation and the ET are given. The techniques are, furthermore, illustrated through examples and compared with other underestimating methods used in global optimization solvers such as BB and BARON.

关键词： signomial functions convex underestimators deterministic global optimization MINLP exponential transformation power transformation

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Solving global optimization problems using reformulations and signomial transformations

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COMPUTERS & CHEMICAL ENGINEERING 2018年 116卷 122-134页

作者： Lundell, A. Westerlund, T. Abo Akad Univ Math & Stat Tuomiokirkontori 1 FIN-20500 Turku Finland Abo Akad Univ Proc Design & Syst Engn Piispankatu 8 FIN-20500 Turku Finland

In this paper, a framework for reformulating nonconvex mixed-integer nonlinear programming (MINLP) problems containing twice-differentiable (C-2) functions to convex relaxed form is discussed. To provide flexibility and for utilizing more effective transformation strategies, the twice-differentiable functions can be partitioned into convex, signomial and general nonconvex functions. The latter two can then be convexified using lifting transformations in combination with approximations using piecewise linear functions (PLFs). However, since there are many degrees of freedom in how to select the set of transformations, an optimization-based method is proposed for finding an optimal set. The lifting transformations are based on single-variable power and exponential transformations for signomials. For nonconvex C-2-functions the alpha reformulation (alpha R) technique as well as more generally the method of difference of convex functions can be applied. In the alpha R, the alpha BB convex underestimator can be used. The framework is utilized in the alpha signomial global optimization (alpha SGO) algorithm to find the epsilon-global solution to a nonconvex problem by iteratively updating the approximations provided by the PLFs. The framework can also be used to directly obtain a convex relaxation of any nonconvex MINLP problem of the specified type to a determined accuracy. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Global optimization Nonconvex MINLP Reformulation techniques signomial functions Twice-differentiable nonconvex functions Power transformations Exponential transformations alpha reformulation alpha BB convex underestimator Difference of convex functions

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Solution of security constrained optimal power flow for large-scale power systems by convex transformation techniques and Taylor series

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IET GENERATION TRANSMISSION & DISTRIBUTION 2016年第4期10卷 889-896页

作者： Attarha, Ahmad Amjady, Nima Semnan Univ Dept Elect Engn Semnan Iran

Security constrained optimal power flow (SCOPF) is a key operation function for today's power systems, extensively used, for example, in the dispatching centres. However, SCOPF is a complex optimisation problem for large practical power systems, mostly due to its highly non-convex nature. This study aims at changing highly non-convex SCOPF problem to a convex one. In this way, SCOPF can be solved with much less computation burden and more optimal solutions can be obtained for it by means of commercial solvers. For this purpose, the non-convex terms of SCOPF model (for instance, the terms associated with the valve loading effects of units and AC network modelling) are first reformulated using signomial functions based on Taylor series. Then, the signomial SCOPF formulation becomes convex by means of power transformation techniques. The constructed convex SCOPF formulation can be easily solved to obtain its optimal solution. The proposed solution approach is tested on well-known IEEE 30-bus, 118-bus, 300-bus and the polish 2746-bus test systems and its obtained results are compared with the results of several other SCOPF solution methods and published literature figures.

关键词： valve loading effects large-scale power systems power system security Taylor series nonconvex SCOPF problem Polish 2746-bus test systems security constrained optimal power flow solution dispatching centres Optimisation techniques Power system control IEEE bus test system power transformation techniques AC network modelling concave programming convex programming load flow complex optimisation problem convex transformation techniques signomial functions

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A reformulation framework for global optimization

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JOURNAL OF GLOBAL OPTIMIZATION 2013年第1期57卷 115-141页

作者： Lundell, Andreas Skjal, Anders Westerlund, Tapio Abo Akad Univ Ctr Excellence Optimizat & Syst Engn SF-20500 Turku Finland

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called alpha BB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.

关键词： Global optimization Reformulation technique Convex underestimators Mixed integer nonlinear programming Twice-differentiable functions signomial functions Piecewise linear functions alpha BB-underestimator SGO-algorithm

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Some transformation techniques with applications in global optimization

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JOURNAL OF GLOBAL OPTIMIZATION 2009年第2-3期43卷 391-405页

作者： Lundell, Andreas Westerlund, Joakim Westerlund, Tapio Abo Akad Univ Proc Design Lab SF-20500 Turku Finland

In this paper some transformation techniques, based on power transformations, are discussed. The techniques can be applied to solve optimization problems including signomial functions to global optimality. signomial terms can always be convexified and underestimated using power transformations on the individual variables in the terms. However, often not all variables need to be transformed. A method for minimizing the number of original variables involved in the transformations is, therefore, presented. In order to illustrate how the given method can be integrated into the transformation framework, some mixed integer optimization problems including signomial functions are finally solved to global optimality using the given techniques.

关键词： Transformation and convexification techniques signomial functions Global optimization Mixed integer non-linear programming

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Increasing the Fitting Precision of Implicit Polynomial Curves in 2D Modeling Through signomial Powers

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IEEE ACCESS 2024年 12卷 69909-69921页

作者： Pence, Ihsan Burdur Mehmet Akif Ersoy Univ Dept Software Engn TR-15300 Burdur Turkiye

Today, common algorithms such as spline curves and triangulation methods are used in 2-and 3-dimensional modeling of objects. However, although these methods can express objects well in terms of visuality because they treat objects as a combination of many small parts, they cannot reveal the mathematical expression of the whole object. The most prominent method of expressing objects with a single equation is the implicit polynomial (IP) curves and the 3L method used to realize the more precise effects of these curves. In this study, the 3L method was accepted as the primary method, and signomial functions using real number powers were used to increase the fitting precision in 2D modeling. The artificial bee colony (ABC) algorithm was used to determine signomial terms. Based on the root mean square error, peak signal-to-noise ratio, coefficient of determination, and structural similarity values obtained in experimental studies, modeling success can be increased. Also, due to adding the initial power expression of IP as the population element for ABC, the sensitivity of the current IP model is increased rather than remodeled.

关键词： Mathematical models Polynomials IP networks Shape Fitting Three-dimensional displays Solid modeling Artificial bee colony algorithm signomial functions implicit polynomial curves 3L method artificial bee colony modeling

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COMPUTATIONAL STUDY OF METHODS FOR SOLVING POLYNOMIAL GEOMETRIC PROGRAMS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1976年第2期19卷 233-259页

作者： DINKEL, JJ KOCHENBERGER, GA MCCARL, B PURDUE UNIV DEPT AGR ECONW LAFAYETTEIN 47907 PENN STATE UNIV DEPT MANAGEMENT SCI & ORG BEHAVUNIVERSITY PKPA 16802

A computational comparison of several methods for dealing with polynomial geometric programs is presented. Specifically, we compare the complementary programs of Avriel and Williams (Ref. 1) with the reversed programs and the harmonic programs of Duffin and Peterson (Refs. 2, 3). These methods are used to generate a sequence of posynomial geometric programs which are solved using a dual algorithm.

关键词： computing methods Geometric programming nonlinear programming signomial functions

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Convexity of products of univariate functions and convexification transformations for geometric programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2008年第3期138卷 407-427页

作者： Gounaris, C. E. Floudas, C. A. Princeton Univ Dept Chem Engn Princeton NJ 08544 USA

We investigate the characteristics that have to be possessed by a functional mapping f : R bar right arrow R so that it is suitable to be employed in a variable transformation of the type x -> f(y) in the convexification of posynomials. We study first the bilinear product of univariate functions f(1)(y(1)), f(2)(y(2)) and, based on convexity analysis, we derive sufficient conditions for these two functions so that F-2(y(1), y(2))=f(1)(y(1))f(2)(y(2)) is convex for all (y(1), y(2)) in some box domain. We then prove that these conditions suffice for the general case of products of univariate functions;that is, they are sufficient conditions for every f (i)(y (i) ), i=1,2, ... , n so as F-n (y(1), y(2), ... , y(n) ) = Pi (n)(i=1) f (i) (y (i)) to be convex. In order to address the transformation of variables that are exponentiated to some power kappa not equal 1, we investigate under which further conditions would the function (f)(kappa) be also suitable. The results provide rigorous reasoning on why transformations that have already appeared in the literature, like the exponential or reciprocal, work properly in convexifying posynomial programs. Furthermore, a useful contribution is in devising other transformation schemes that have the potential to work better with a particular formulation. Finally, the results can be used to infer the convexity of multivariate functions that can be expressed as products of univariate factors, through conditions on these factors on an individual basis.

关键词： global optimization geometric programming multilinear products posynomial functions signomial functions convexification transformations

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