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In this paper, we present methods of penalty functions for solving constrained optimization problems. The methods that we characterize presently, attempt to approximate a constrained optimization problem with an unconstrained, one and then apply standard search techniques such as the exterior penalty function method and the interior penalty method to get solutions. The paper that follows assure exterior penalty methods recognizing that interior penalty function methods incarnate the same principles.

Despite the wide adoption of spike-and-slab methodology for Bayesian variable selection, its potential for penalized likelihood estimation has largely been overlooked. In this article, we bridge this gap by cross-fertilizing these two paradigms with the Spike-and-Slab LASSO procedure for variable selection and parameter estimation in linear regression. We introduce a new class of self-adaptive penalty functions that arise from a fully Bayes spike-and-slab formulation, ultimately moving beyond the separable penalty framework. A virtue of these nonseparable penalties is their ability to borrow strength across coordinates, adapt to ensemble sparsity information and exert multiplicity adjustment. The Spike-and-Slab LASSO procedure harvests efficient coordinate-wise implementations with a path-following scheme for dynamic posterior exploration. We show on simulated data that the fully Bayes penalty mimics oracle performance, providing a viable alternative to cross-validation. We develop theory for the separable and nonseparable variants of the penalty, showing rate-optimality of the global mode as well as optimal posterior concentration when p > n. Supplementary materials for this article are available online.

This article develops a new penalty‐based aggregation operator known as the penalty‐based induced ordered weighted averaging (P‐IOWA) operator which is an extension of penalty‐based ordered weighted averaging (P‐OWA) operator. Our goal is to figure out how the induced variable realigns penalties when gathering information. We extend the P‐OWA and P‐IOWA operators with the different means such as generalized mean and quasi‐arithmetic mean. This article also includes different families of P‐OWA and P‐IOWA operators. The value of these new operators is demonstrated through a case study centered on investment matters. This study evaluates the economic and governance performance of seven South Asian nations utilizing nine indicators from 2021 data. The research examines 5 economic indicators including GDP growth, exports and imports (% of GDP), inflation, and labor force metrics, alongside 4 governance indicators focusing on corruption control, government effectiveness, and political stability. We use min–max normalization to standardize the varied values, which originally ranged from 0.5% to 77.7% across various metrics. Following this, the normalized inverse penalty method is used to derive optimal weights for these indicators, tackling the task of combining multidimensional data. Subsequently, we implement and evaluate various penalty‐based aggregation methodologies on the normalized data, each offering a distinct approach to penalizing outliers and balancing indicator weights. The study compares the results obtained from these operators to assess their impact on country rankings and overall performance evaluation. This approach allows for a comprehensive comparison of countries’ performances, integrating both economic and governance dimensions into a single, quantifiable framework.

Drawing upon rational choice and investor attention theories, we examine how accusations of corporate bribery and subsequent investigations shape market reactions. Using event study methodology to measure loss in firm value for public firms facing bribery investigations from 1978 to 2010, we found that total market penalties amounted to $60.61 billion. We ran moderated multiple regression analysis to examine further the degree to which the unique characteristics of bribery explain variations in market penalties. Companies committing bribery in less corrupt host countries and with the involvement of compromised executives experienced greater market penalties than did other companies. After partitioning share value losses into components for regulatory penalties, class action settlements, and loss to reputation, we found that reputational penalties account for 81.8c̷ of every dollar of share value loss. Omission of reputational penalties in rational choice calculus underestimates bribery costs by 4.5 times. The results suggest that firms should not underestimate the importance of market-imposed reputational penalties by merely considering regulator-imposed fines and sanctions.

Optimal stomatal control models have shown great potential in predicting stomatal behavior and improving carbon cycle modeling. Basic stomatal optimality theory posits that stomatal regulation maximizes the carbon gain relative to a penalty of stomatal opening. All models take a similar approach to calculate instantaneous carbon gain from stomatal opening (the gain function). Where the models diverge is in how they calculate the corresponding penalty (the penalty function). In this review, we compare and evaluate 10 different optimization models in howthey quantify the penalty andhowwell they predict stomatal responses to the environment. Weevaluate models in two ways. First, we compare their penalty functions against seven criteria that ensure a unique and qualitatively realistic solution. Second, we quantitatively test model against multiple leaf gas-exchange datasets. The optimization models with better predictive skills have penalty functions that meet our seven criteria and use fitting parameters that are both few in number and physiology based. The most skilled models are those with a penalty function based on stress-induced hydraulic failure. We conclude by proposing a new model that has a hydraulics-based penalty function that meets all seven criteria and demonstrates a highly predictive skill against our test datasets.

It has been increasingly reported that the multiobjective optimization evolutionary algorithm based on decomposition (MOEA/D) is promising for handling multiobjective optimization problems (MOPs). MOEA/D employs scalarizing functions to convert an MOP into a number of single-objective subproblems. Among them, penalty boundary intersection (PBI) is one of the most popular decomposition approaches and has been widely adopted for dealing with MOPs. However, the original PBI uses a constant penalty value for all subproblems and has difficulties in achieving a good distribution and coverage of the Pareto front for some problems. In this paper, we investigate the influence of the penalty factor on PBI, and suggest two new penalty schemes, i.e., adaptive penalty scheme and subproblem-based penalty scheme (SPS), to enhance the spread of Pareto-optimal solutions. The new penalty schemes are examined on several complex MOPs, showing that PBI with the use of them is able to provide a better approximation of the Pareto front than the original one. The SPS is further integrated into two recently developed MOEA/D variants to help balance the population diversity and convergence. Experimental results show that it can significantly enhance the algorithm’s performance.

Optimization methods play a crucial role in various fields and applications. In some optimization problems, the derivative information of the objective function is unavailable. Such black-box optimization problems need to be solved by derivative-free optimization methods. At the same time, optimization problems with ellipsoidal constraints are important and have widespread applications in various fields as well. Following the development of the late professor M. J. D. Powell’s efficient derivative-free trust-region optimization methods, this paper considers solving derivative-free optimization problems on the ellipsoid. Our new optimization solver EC-NEWUOA for problems on the ellipsoid in ℜ n is designed based on Powell’s derivative-free software NEWUOA for unconstrained optimization problems. The proposed techniques for our new method mainly include using the Courant penalty function, the augmented Lagrangian method, and the projection technique. Details about the method and theoretical analysis are included in this paper. We also compare our new method with other algorithms by solving test problems and then show the numerical advantages of our new method.

Background Multiple co-inertia analysis (mCIA) is a multivariate analysis method that can assess relationships and trends in multiple datasets. Recently it has been used for integrative analysis of multiple high-dimensional -omics datasets. However, its estimated loading vectors are non-sparse, which presents challenges for identifying important features and interpreting analysis results. We propose two new mCIA methods: 1) a sparse mCIA method that produces sparse loading estimates and 2) a structured sparse mCIA method that further enables incorporation of structural information among variables such as those from functional genomics. Results Our extensive simulation studies demonstrate the superior performance of the sparse mCIA and structured sparse mCIA methods compared to the existing mCIA in terms of feature selection and estimation accuracy. Application to the integrative analysis of transcriptomics data and proteomics data from a cancer study identified biomarkers that are suggested in the literature related with cancer disease. Conclusion Proposed sparse mCIA achieves simultaneous model estimation and feature selection and yields analysis results that are more interpretable than the existing mCIA. Furthermore, proposed structured sparse mCIA can effectively incorporate prior network information among genes, resulting in improved feature selection and enhanced interpretability.

This article develops a framework for analyzing the risks of poverty in terms of prevalences (share of the population with a risk) and penalties (increased probability of poverty associated with a risk). A comparison of the four major risks (low education, single motherhood, young headship, and unemployment) across 29 rich democracies reveals there is greater variation in penalties than prevalences. The United States has high poverty partly because it has the highest penalties despite below average prevalences. Also, U.S. poverty in 2013 would be worse with prevalences from 1970 or 1980. There is little evidence that penalties discourage prevalences, while welfare generosity significantly moderates the penalties for unemployment and low education. The authors conclude that a focus on risks does not provide a convincing explanation of poverty, single motherhood may be the least important of the risks, and studies based solely on the United States are constrained by potentially large sample selection biases.

In this paper, a class of smooth penalty functions is proposed for constrained optimization problem. It is put forward based on L p , a smooth function of a class of exact penalty function ℓ p p ∈ ( 0 , 1 ] . Based on the class of penalty functions, a penalty algorithm is presented. Under the very weak condition, a perturbation theorem is set up. The global convergence of the algorithm is derived. This result generalizes some existing conclusions. Finally, numerical experiments on two examples demonstrate the effectiveness and efficiency of our algorithm.