Penalty Problem Research Articles

Well-Posedness and Convergence Results for History-Dependent Inclusions

We consider an inclusion in a real Hilbert space governed by a time-dependent set of constraints and a history-dependent operator. We introduce the concept of T - well-posedness for this inclusion, associated to a given Tykhonov triple T . Next, we provide a T -well-posedness result that we use in order to deduce the continuous dependence of the solution with respect to the data. Then, we state and prove a convergence criterion to the solution of the inclusion that we use to prove a convergence result for an associate penalty problem. Moreover, we show that this criterion allows us to construct a Tykhonov triple T -which give rise to an optimal well-posedness concept for the corresponding inclusion. Finally, we use these abstract results in the study of a nonlinear viscoelastic constitutive law with long memory term and unilateral constraints.

The mixed penalty method for the Signorini problem

We investigate two penalty approaches for the Signorini problem in the mixed form. The well-posedness theory has been established for the two mixed penalty problems in both the continuous and discrete sense. For the continuous case, we obtain the error of the penalty for both two approaches. We also study the error estimates of the discrete mixed penalty problems, which depend on the mesh size and the penalty parameter. Based on the two penalty approaches, we design two algorithms and show their convergence. Several numerical experiments are carried out to confirm the theoretical results.

Optimized Decentralized Reward Distribution(1)

In DeFi (Decentralized Finance) applications, and in dApps (Decentralized Application) generally, it is common to periodically pay interest to users as an incentive, or periodically collect a penalty from them as a deterrent. If we view the penalty as a negative reward, both the interest and penalty problems come down to the problem of distributing rewards. Reward distribution is quite accomplishable in financial management where general computers are used, but on a blockchain, where computational resources are inherently expensive and the amount of computation per transaction is absolutely limited with a predefined, uniform quota, not only do the system administrators have to pay heavy gas fees if they handle rewards of many users one by one, but the transaction may also be terminated on the way. The computational quota makes it impossible to guarantee processing an unknown number of users. We propose novel algorithms that solve Simple Interest, Simple Burn, Compound Interest, and Compound Burn tasks, which are typical components of DeFi applications. If we put numerical errors aside, these algorithms realize accurate distribution of rewards to an unknown number of users with no approximation, while adhering to the computational quota per transaction. For those who might already be using similar algorithms, we prove the algorithms rigorously so that they can be transparently presented to users. We also introduce reusable concepts and notations in decentralized reasoning, and demonstrate how they can be efficiently used. We demonstrate, through simulated tests spanning over 128 simulated years, that the numerical errors do not grow to a dangerous level.

What Goes Around Comes Around: Covering Tours and Cycle Covers with Turn Costs

We investigate several geometric problems of finding tours and cycle covers with minimum turn cost, which have been studied in the past, with complexity, approximation results, and open problems dating back to work by Arkin et al. in 2001. Many new practical applications have spawned variants: For full coverage, all points have to be covered, for subset coverage, specific points have to be covered, and for penalty coverage, points may be left uncovered by incurring a penalty. We show that finding a minimum-turn (full) cycle cover is NP-hard even in 2-dimensional grid graphs, solving the long-standing open Problem 53 in The Open Problems Project edited by Demaine, Mitchell and O’Rourke. We also prove NP-hardness of finding a subset cycle cover of minimum turn cost in thin grid graphs, for which Arkin et al. gave a polynomial-time algorithm for full coverage; this shows that their boundary techniques cannot be applied to compute exact solutions for subset and penalty variants. We also provide a number of positive results. In particular, we establish the first constant-factor approximation algorithms for all considered subset and penalty problem variants for grid-based instances, based on LP/IP techniques. These geometric versions allow many possible edge directions (and thus, turn angles, such as in hexagonal grids or higher-dimensional variants); our approximation factors improve the combinatorial ones of Arkin et al.

A Convergence Criterion for a Class of Stationary Inclusions in Hilbert Spaces

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f∈X. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}⊂X, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws.

Robust and Sparse Portfolio: Optimization Models and Algorithms

The robust and sparse portfolio selection problem is one of the most-popular and -frequently studied problems in the optimization and financial literature. By considering the uncertainty of the parameters, the goal is to construct a sparse portfolio with low volatility and decent returns, subject to other investment constraints. In this paper, we propose a new portfolio selection model, which considers the perturbation in the asset return matrix and the parameter uncertainty in the expected asset return. We define three types of stationary points of the penalty problem: the Karush–Kuhn–Tucker point, the strong Karush–Kuhn–Tucker point, and the partial minimizer. We analyze the relationship between these stationary points and the local/global minimizer of the penalty model under mild conditions. We design a penalty alternating-direction method to obtain the solutions. Compared with several existing portfolio models on seven real-world datasets, extensive numerical experiments demonstrate the robustness and effectiveness of our model in generating lower volatility.

A Regularized Tseng Method for Solving Various Variational Inclusion Problems and Its Application to a Statistical Learning Model

We study three classes of variational inclusion problems in the framework of a real Hilbert space and propose a simple modification of Tseng’s forward-backward-forward splitting method for solving such problems. Our algorithm is obtained via a certain regularization procedure and uses self-adaptive step sizes. We show that the approximating sequences generated by our algorithm converge strongly to a solution of the problems under suitable assumptions on the regularization parameters. Furthermore, we apply our results to an elastic net penalty problem in statistical learning theory and to split feasibility problems. Moreover, we illustrate the usefulness and effectiveness of our algorithm by using numerical examples in comparison with some existing relevant algorithms that can be found in the literature.

MODELING AND NUMERICAL SIMULATION OF THE PENALTY METHOD FOR UNILATERAL CONTACT PROBLEM WITH TRESCA’S FRICTION IN THERMO-ELECTRO-VISCO-ELASTICITY

In this work, we consider the penalty method applied to contact problem in thermo-electro-visco-elasticity with Signorini’s condition and Tresca’s friction law. Mathe- matical properties, such as the existence of a solution to the penalty problem and its conver- gence to the solution of the original problem, are reported. Then, we present some numerical results in the study of a two-dimensional test problem and we establish its convergence.

Double inertial parameters forward-backward splitting method: Applications to compressed sensing, image processing, and SCAD penalty problems

Double inertial parameters forward-backward splitting method: Applications to compressed sensing, image processing, and SCAD penalty problems

Fourier-spectral method for the Landau–Lifshitz–Gilbert equation in micromagnetism

The Landau–Lifshitz–Gilbert (LLG) equation models the temporal evolution of magnetization in continuum ferromagnets. The LLG equation has a nonconvex constraint and is highly nonlinear. In this paper, we will use the Fourier-spectral method for approximating the solution of the LLG equation with the nonconvex constraint. We consider the penalty problem and show the stability and convergence of the approximate penalty problem, and then we show the convergence of the penalty problem to a (weak) solution of the LLG equation. Computational experiments and comparison with other numerical methods are presented to demonstrate the effectiveness of the proposed method.

The exactness of the [formula omitted] penalty function for a class of mathematical programs with generalized complementarity constraints

In a Mathematical Program with Generalized Complementarity Constraints (MPGCC), complementarity relation is imposed between each pair of variable blocks. MPGCC includes the traditional Mathematical Program with Complementarity Constraints (MPCC) as a special case. On account of the disjunctive feasible region, MPCC and MPGCC are generally difficult to handle. The ℓ1 penalty method, often adopted in computation, opens a way of circumventing the difficulty. Yet it remains unclear about the exactness of the ℓ1 penalty function, namely, whether there exists a sufficiently large penalty parameter so that the penalty problem shares the optimal solution set with the original one. In this paper, we consider a class of MPGCCs that are of multi-affine objective functions. This problem class finds applications in various fields, e.g., the multi-marginal optimal transport problems in many-body quantum physics and the pricing problems in network transportation. We first provide an instance from this class, the exactness of whose ℓ1 penalty function cannot be derived by existing tools. We then establish the exactness results under rather mild conditions. Our results cover those existing ones for MPCC and apply to multi-block contexts.

An exact penalty function approach for inequality constrained optimization problems based on a new smoothing technique

Exact penalty methods are one of the effective tools to solve nonlinear programming problems with inequality constraints. In this study, a new class of exact penalty functions is defined and a new family of smoothing techniques to exact penalty functions is introduced. Error estimations are presented among the original, non-smooth exact penalty and smoothed exact penalty problems. It is proved that an optimal solution of smoothed penalty problem is an optimal solution of original problem. A smoothing penalty algorithm based on the the new smoothing technique is proposed and the convergence of the algorithm is discussed. Finally, the efficiency of the algorithm on some numerical examples is illustrated.

Penalty method for solving a class of stochastic differential variational inequalities with an application

The aim of this paper is to study the penalty method for solving a class of stochastic differential variational inequalities (SDVIs). The penalty problem for solving SDVIs is first constructed and the convergence of the sequences generated by the penalty problem is proved under some mild conditions. As an application, the convergence of the sequences generated by the penalty problem is obtained for solving a stochastic migration equilibrium problem with movement cost.

Error bound and exact penalty method for optimization problems with nonnegative orthogonal constraint

Abstract This paper is concerned with a class of optimization problems with the non-negative orthogonal constraint, in which the objective function is $L$-smooth on an open set containing the Stiefel manifold $\textrm {St}(n,r)$. We derive a locally Lipschitzian error bound for the feasible points without zero rows when $n>r>1$, and when $n>r=1$ or $n=r$ achieve a global Lipschitzian error bound. Then, we show that the penalty problem induced by the elementwise $\ell _1$-norm distance to the non-negative cone is a global exact penalty, and so is the one induced by its Moreau envelope under a lower second-order calmness of the objective function. A practical penalty algorithm is developed by solving approximately a series of smooth penalty problems with a retraction-based nonmonotone line-search proximal gradient method, and any cluster point of the generated sequence is shown to be a stationary point of the original problem. Numerical comparisons with the ALM [Wen, Z. W. & Yin, W. T. (2013, A feasible method for optimization with orthogonality constraints. Math. Programming, 142, 397–434),] and the exact penalty method [Jiang, B., Meng, X., Wen, Z. W. & Chen, X. J. (2022, An exact penalty approach for optimization with nonnegative orthogonality constraints. Math. Programming. https://doi.org/10.1007/s10107-022-01794-8)] indicate that our penalty method has an advantage in terms of the quality of solutions despite taking a little more time.

Study of weak solutions for degenerate parabolic inequalities with nonstandard conditions

In this paper, we study the degenerate parabolic variational inequalities in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.

Chance constrained dynamic optimization approach for single machine scheduling involving flexible maintenance, production, and uncertainty

Chance constraints are suitable for industrial process modeling under uncertain conditions, where constraints cannot be strictly satisfied or do not need to be fully satisfied. In this paper, a single machine scheduling problem involving flexible maintenance, production, and uncertainty is modeled as a chance constrained dynamic optimization problem (CCDOP). A novel method is proposed for transforming the CCDOP into an equivalent deterministic dynamic optimization problem (DOP) with fixed state jump times. Furthermore, by using the idea of l1 penalty function and a smooth approximation technique, the resulting deterministic DOP becomes a smoothing penalty problem, which is a non-convex nonlinear parameter optimization problem (NNPOP) with simple bounds on the variables. To solve the NNPOP, a gradient-based stochastic search algorithm (GSSA) is developed based on a gradient-based adaptive search algorithm (GASA) and a novel stochastic search algorithm (NSSA). The convergence analysis result shows that the GSSA is a globally convergent algorithm. Finally, two numerical examples are used to illustrate the effectiveness of the proposed method. Numerical results show that the GSSA has excellent convergence behavior with robust computation feature, providing better results compared with the other typical methods.

1-Norm random vector functional link networks for classification problems

This paper presents a novel random vector functional link (RVFL) formulation called the 1-norm RVFL (1N RVFL) networks, for solving the binary classification problems. The solution to the optimization problem of 1N RVFL is obtained by solving its exterior dual penalty problem using a Newton technique. The 1-norm makes the model robust and delivers sparse outputs, which is the fundamental advantage of this model. The sparse output indicates that most of the elements in the output matrix are zero; hence, the decision function can be achieved by incorporating lesser hidden nodes compared to the conventional RVFL model. 1N RVFL produces a classifier that is based on a smaller number of input features. To put it another way, this method will suppress the neurons in the hidden layer. Statistical analyses have been carried out on several real-world benchmark datasets. The proposed 1N RVFL with two activation functions viz., ReLU and sine are used in this work. The classification accuracies of 1N RVFL are compared with the extreme learning machine (ELM), kernel ridge regression (KRR), RVFL, kernel RVFL (K-RVFL) and generalized Lagrangian twin RVFL (GLTRVFL) networks. The experimental results with comparable or better accuracy indicate the effectiveness and usability of 1N RVFL for solving binary classification problems.

Study of weak solutions of variational inequality systems with degenerate parabolic operators and quasilinear terms arising Americian option pricing problems

<abstract><p>In this paper, we study variational inequality systems with quasilinear degenerate parabolic operators in a bounded domain. As a series of penalty problems, the existence of the solutions in the weak sense is proved by a limit process. The uniqueness of the solution is also proved.</p></abstract>

Обзор методов пространственной верификации и их применение для ансамблевых прогнозов

State-of-the-art high-resolution NWP models simulate mesoscale systems with a high degree of detail, with large amplitudes and high gradients of fields of weather variables. Higher resolution leads to the spatial and temporal error growth and to a well-known double penalty problem. To solve this problem, the spatial verification methods have been developed over the last two decades, which ignore moderate errors (especially in the position), but can still evaluate the useful skill of a high-resolution model. The paper refers to the updated classification of spatial verification methods, briefly describes the main methods, and gives an overview of the international projects for intercomparison of the methods. Special attention is given to the application of the spatial approach to ensemble forecasting. Popular software packages are considered. The Russian translation is proposed for the relevant English terms. Keywords: high-resolution models, verification, double penalty, spatial methods, ensemble forecasting, object-based methods

Convergence analysis of modified inertial forward–backward splitting scheme with applications

In the settings of real Hilbert spaces, this paper proposes modified forward– backward scheme with inertial extrapolation step for solving inverse problems. In our proposed method, it is possible for the inertial factor to be chosen as 1 unlike many previous inertial forward–backward splitting methods available in the literature. Weak and strong convergence of our proposed method are obtained under some standard conditions. In order to show the numerical advantage gained when the inertial factor is chosen as 1, a number of numerical implementations are performed on two different inverse problems: LASSO and SCAD penalty problems.