Classification of some penalty methods

Optimization problems arise in science, engineering, economy, etc. and we need to find the best solutions for each reality. The methods used to solve these problems depend on several factors, including the amount and type of accessible information, the available algorithms for solving them, and, obviously, the intrinsic characteristics of the problem. There are many kinds of optimization problems and, consequently, many kinds of methods to solve them. When the involved functions are nonlinear and their derivatives are not known or are very difficult to calculate, these methods are more rare. These kinds of functions are frequently called black box functions. To solve such problems without constraints (unconstrained optimization), we can use direct search methods. These methods do not require any derivatives or approximations of them. But when the problem has constraints (nonlinear programming problems) and, additionally, the constraint functions are black box functions, it is much more difficult to find the most appropriate method. Penalty methods can then be used. They transform the original problem into a sequence of other problems, derived from the initial, all without constraints. Then this sequence of problems (without constraints) can be solved using the methods available for unconstrained optimization. In this chapter, we present a classification of some of the existing penalty methods and describe some of their assumptions and limitations. These methods allow the solving of optimization problems with continuous, discrete, and mixing constraints, without requiring continuity, differentiability, or convexity. Thus, penalty methods can be used as the first step in the resolution of constrained problems, by means of methods that typically are used by unconstrained problems. We also discuss a new class of penalty methods for nonlinear optimization, which adjust the penalty parameter dynamically.

Penalty fuzzy function for derivative-free optimization

Penalty and Barrier methods are normally used to solve Nonlinear Optimization Problems constrained problems. The problems appear in areas such as engineering and are often characterised by the fact that involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. This means that optimization methods based on derivatives cannot net used. A Java based API was implemented, including only derivative-free optimizationmethods, to solve both constrained and unconstrained problems, which includes Penalty and Barriers methods. In this work a new penalty function, based on Fuzzy Logic, is presented. This function imposes a progressive penalization to solutions that violate the constraints. This means that the function imposes a low penalization when the violation of the constraints is low and a heavy penalisation when the violation is high. The value of the penalization is not known in beforehand, it is the outcome of a fuzzy inference engine. Numerical results comparing the proposed function with two of the classic penalty/barrier functions are presented. Regarding the presented results one can conclude that the prosed penalty function besides being very robust also exhibits a very good performance.

Direct-search penalty/barrier methods

In Nonlinear Optimization Penalty and Barrier Methods are normally used to solve Constrained Problems. There are several Penalty/Barrier Methods and they are used in several areas from Engineering to Economy, through Biology, Chemistry, Physics among others. In these areas it often appears Optimization Problems in which the involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. In this work some Penalty/Barrier functions are tested and compared, using in the internal process, Derivative-free, namely Direct Search, methods. This work is a part of a bigger project involving the development of an Application Programming Interface, that implements several Optimization Methods, to be used in applications that need to solve constrained and/or unconstrained Nonlinear Optimization Problems. Besides the use of it in applied mathematics research it is also to be used in engineering software packages.

Adaptive Penalty and Barrier function based on Fuzzy Logic

Optimization methods have been used in many areas of knowledge, such as Engineering, Statistics, Chemistry, among others, to solve optimization problems. In many cases it is not possible to use derivative methods, due to the characteristics of the problem to be solved and/or its constraints, for example if the involved functions are non-smooth and/or their derivatives are not know. To solve this type of problems a Java based API has been implemented, which includes only derivative-free optimization methods, and that can be used to solve both constrained and unconstrained problems. For solving constrained problems, the classic Penalty and Barrier functions were included in the API. In this paper a new approach to Penalty and Barrier functions, based on Fuzzy Logic, is proposed. Two penalty functions, that impose a progressive penalization to solutions that violate the constraints, are discussed. The implemented functions impose a low penalization when the violation of the constraints is low and a heavy penalty when the violation is high. Numerical results, obtained using twenty-eight test problems, comparing the proposed Fuzzy Logic based functions to six of the classic Penalty and Barrier functions are presented. Considering the achieved results, it can be concluded that the proposed penalty functions besides being very robust also have a very good performance.

A firefly dynamic penalty approach for solving engineering design problems

Firefly Algorithm is a recent swarm intelligence method, inspired by the social behavior of fireflies, based on their flashing and attraction characteristics [1, 2]. In this paper, we analyze the implementation of a dynamic penalty approach combined with the Firefly algorithm for solving constrained global optimization problems. In order to assess the applicability and performance of the proposed method, some benchmark problems from engineering design optimization are considered.

The Part-time pay penalty in a segmented labor market

This paper is the first to examine the implications of switching to PT work for women's subsequent earnings trajectories, distinguishing by their type of contract: permanent or fixedterm. Using a rich longitudinal Spanish data set from Social Security records of over 76,000 prime-aged women strongly attached to the Spanish labor market, we find that PT work aggravates the segmentation of the labor market insofar there is a PT pay penalty and this penalty is larger and more persistent in the case of women with fixed-term contracts. The paper discusses problems arising in empirical estimation (including a problem not discussed in the literature up to now: the differential measurement error of the LHS variable by PT status), and how to address them. It concludes with policy implications relevant for Continental Europe and its dual structure of employment protection.

A direct approach to the discounted penalty function

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1 784 (2010)], which uses different methods.

Is there a wage penalty for horizontal and vertical mismatch?

This paper studies how the horizontal and vertical mismatches in the labor market affect wage. We do so by taking into account that by choosing a job, wage and mismatches are simultaneously determined. The Seemingly Unrelated Equations model also allows us to control for any omitted variable that could cause biased estimators. We use REFLEX data for Spain. Results reveal that in most cases being horizontally matched has a wage premium and being over-educated does not affect wage. Results suggest that the modeling strategy successfully accounts for some omitted variable that affects simultaneously the probability of being horizontally matched and the wage. This could explain the existence of a wage penalty for over-educated workers when the omitted variable issue is not dealt with.

La importància de les accions a pilota aturada en el futbol professional

L’objectiu del present estudi va consistir en estudiar el percentatge de gols a pilota aturada de la Lliga BBVA i la Premier League, establint el percentatge d’efectivitat de cadascuna d’elles. A més, es volia analitzar la freqüència de les diferents tipus d’accions a pilota aturada i substituir el terme estratègia per accions a pilota aturada. Amb aquest objectiu es van analitzar 80 partits disputats per equips de futbol professional en la temporada 2012-13 de la Lliga BBVA i la Premier League mitjançant l’anàlisi de vídeos. Les variable dependent era la consecució o no del gol, tenint com a variables independents 4 tipus d’accions a pilota aturada: tirs lliures directes, tirs lliures indirectes, córners i penals. Els resultats indiquen un major número de gols provinents d’accions a pilota aturada en la Lliga BBVA que en la Premier League, 29,8% per un 23,4% respectivament. Tot i això, s’ha comprovat com els percentatges d’efectivitat són pràcticament idèntics: 4,54% a la Lliga BBVA i 4,58% a la Premier League. Finalment, s’ha identificat el córner com l’acció que més es produeix en el joc, seguit de les faltes directes, els penals i les faltes indirectes.

Overeducation, skills and wage penalty: Evidence for Spain using PIAAC data

The literature on educational mismatches finds that overeducated workers suffer a wage penalty compared with properly educated workers with the same level of education. Recent literature also suggests that individuals’ skill heterogeneity could explain wage differences between overeducated and properly matched workers. The hypothesis is that overeducated workers earn less due to their lower competences and skills in relative terms. However, that hypothesis has been rarely tested due to data limitations on individuals’ skills. The aim of this paper is to test the individuals’ skill heterogeneity theory in Spain using microdata from PIAAC, because it is one of the developed countries supporting the highest overeducation rates and where its adult population holds the lowest level of skills among a set of developed countries. Our hypothesis is that the wage penalty of overeducation in Spain is explained by the lower skill level of overeducated workers. The obtained evidence confirms this hypothesis but only to a certain extent as skills only explain partially the wage penalty of overeducation.

Is there a motherhood penalty? decomposing the family wage gap in Colombia

The aim of this paper is to provide an estimation and decomposition of the motherhood wage penalty in Colombia. Our empirical strategy was based on the matching procedure designed by Ñopo (The Review of Economics and Statistics, 90(2), 290–299, 2008a ) for the case of gender wage gaps. This is an alternative procedure to the well-known Blinder–Oaxaca decomposition method. Using the cross-sectional data of the Colombian Living Standard Survey, the wage gap was decomposed into four components, according to the characteristics of mothers and non-mothers. Three of the components are explained by differences in observable characteristics of women, while the other is the unexplained part of the gap. We found that mothers earn, on average, 1.73 % less than their counterparts without children and that this gap slightly decreased when the group included older women. It is observed from the results that, once schooling was included as a matching variable, the unexplained part of the gap considerably decreased and became non-significant. Thus, we did not find evidence of wage discrimination against mothers in the Colombian labor market. Copyright Springer Science+Business Media New York 2013

An evaluation of the costs of making specific secondary metabolites: does the yield penalty incurred by host plant resistance to insects due to competition for resources?

The presumption that the synthesis of 'defence' compounds in plants must incur some 'trade-off' or penalty in terms of annual crop yields has been used to explain observed inverse correlations between resistance to herbivores and rates of growth or photosynthesis. An analysis of the cost of making secondary compounds suggests that this accounts for only a small part of the overall carbon budget of annual crop plants. Even the highest reported amounts of secondary metabolites found in different crop species (flavonoids, allylisothiocyanates, hydroxamic acids, 2-tridecanone) represent a carbon demand that can be satisfied by less than an hour's photosynthesis. Similar considerations apply to secondary compounds containing nitrogen or sulphur, which are unlikely to represent a major investment compared to the cost of making proteins, the major demand for these elements. Decreases in growth and photosynthesis in response to stress are more likely the result of programmed down-regulation. Observed correlations between yield and low contents of unpalatable or toxic compounds may be the result of parallel selection during the refinement of crop species by humans.