Exponential Penalty Research Articles

A smoothing iterative method for the finite minimax problem

In this paper, we proposed a smoothing iterative method to solve the finite minimax problems based on the exponential penalty function of Kort and Bertsekas (1972). This approach can be viewed as the extension of one conjugate gradient method. Under suitable conditions, the proposed method is globally convergent. Preliminary numerical results and comparisons show that the proposed method is effective and promising.

Frequency-Domain Computation of Quadratic-Exponential Cost Functionals for Linear Quantum Stochastic Systems

This paper is concerned with quadratic-exponential functionals (QEFs) as risk-sensitive performance criteria for linear quantum stochastic systems driven by multichannel bosonic fields. Such costs impose an exponential penalty on quadratic functions of the quantum system variables over a bounded time interval, and their minimization secures a number of robustness properties for the system. We use an integral operator representation of the QEF, obtained recently, in order to compute its infinite-horizon asymptotic growth rate in the invariant Gaussian state when the stable system is driven by vacuum input fields. The resulting frequency-domain formula expresses the QEF growth rate in terms of two spectral functions associated with the real and imaginary parts of the quantum covariance kernel of the system variables. We also discuss the computation of the QEF growth rate using homotopy and contour integration techniques and provide an illustrative numerical example with a two-mode open quantum harmonic oscillator.

Closed-Form Analysis of Non-Linear Age of Information in Status Updates With an Energy Harvesting Transmitter

Timely status updates are crucial to enabling applications in the massive Internet of Things (IoT). This paper measures the data-freshness performance of a status update system with an energy-harvesting transmitter, considering the randomness in information generation, transmission, and energy harvesting. The performance is evaluated by a non-linear function of age of information (AoI) that is defined as the time elapsed since the generation of the most up-to-date status information at the receiver. The system is formulated as two queues with status packet generation and energy arrivals both assumed to be Poisson processes. With negligible service time, both first-come-first-served (FCFS) and last-come-first-served (LCFS) disciplines for arbitrary buffer and battery capacities are considered, and a method for calculating the average penalty with non-linear penalty functions is proposed. The average AoI, the average penalty under exponential penalty function, and the AoI’s threshold violation probability are obtained in a closed form. When the service time is assumed to follow exponential distribution, a matrix geometric method is used to obtain the average peak AoI. The results illustrate that under the FCFS discipline, the status update frequency needs to be carefully chosen according to the service rate and energy arrival rate in order to minimize the average penalty.

A second-order convergence augmented Lagrangian method using non-quadratic penalty functions

The purpose of the present paper is to study the global convergence of a practical Augmented Lagrangian model algorithm that considers non-quadratic Penalty–Lagrangian functions. We analyze the convergence of the model algorithm to points that satisfy the Karush–Kuhn–Tucker conditions and also the weak second-order necessary optimality condition. The generation scheme of the Penalty–Lagrangian functions includes the exponential penalty function and the logarithmic-barrier without using convex information.

Socially Aware Dynamic Computation Offloading Scheme for Fog Computing System With Energy Harvesting Devices

Fog computing is considered as a promising technology to meet the ever-increasing computation requests from a wide variety of mobile applications. By offloading the computation-intensive requests to the fog node or the central cloud, the performance of the applications, such as energy consumption and delay, are able to be significantly enhanced. Meanwhile, utilizing the recent advances of social network and energy harvesting (EH) techniques, the system performance could be further improved. In this paper, we take the social relationships of the EH mobile devices (MDs) into the design of computational offloading scheme in fog computing. With the objective to minimize the social group execution cost, we advocate game theoretic approach and propose a dynamic computation offloading scheme designing the offloading process in fog computing system with EH MDs. Different queue models are applied to model the energy cost and delay performance. It can be seen that the proposed problem can be formulated as a generalized Nash equilibrium problem (GNEP) and we can use exponential penalty function method to transform the original GNEP into a classical Nash equilibrium problem and address it with semi-smooth Newton method with Armijo line search. The simulation results demonstrate the effectiveness of the proposed scheme.

A novel hybrid trust region minimax fitting algorithm for accurate dimensional metrology of aspherical shapes

Ultra-high precision measuring machines enable to measure aspheric shapes with an uncertainty of few tens of nanometres. The resulting clouds of points are then associated to theoretical model at the same level of accuracy so as to obtain parameters that indicate about form error. Minimum zone (MZ), defined as the least value of peak to valley (PV), is widely used to assess form error. Least squares method (L2) is often used to determine MZ but the resulting value is usually overestimated. For this reason, L2 is replaced by L∞ norm because it gives a more accurate value of MZ since it directly minimizes PV. Using L∞ norm results in a non-smooth optimization problem and consequently its resolution becomes more challenging compared to L2.In this paper, a novel minimax fitting method for accurate metrology of aspheres and freeform based on a hybrid trust region algorithm (HTR) is proposed. To assess performance of the introduced method, it was compared to an available minimax fitting algorithm based on a smoothing technique: exponential penalty function (EPF). The choice of EPF is justified by superior performances in comparison to existing techniques. Comparison was conducted on reference data, data available in literature and data gathered form measurements of a real optical high quality asphere. Results show superiority of HTR over EPF in both returned PV values and execution time.

Convergence of Exponential Penalty Function Method for Variational Problems

In this paper, we propose an exponential penalty function method in order to solve a constrained variational problem by transforming it into a sequence of unconstrained ones. Further, we analyze the relationship between the optimal solutions of the sequence of exponential penalized variational problems and that of the original constrained variational problem. The convergence of this exponential penalty method is also examined for variational problems. Numerical examples are provided to verify the obtained results and validate the efficient use of exponential penalty method for solving constrained variational problems.

Investigation of minimum zone assessment methods for aspheric shapes

Improvement in metrology of asphere and freeform surfaces is requested in many sectors and industries. ISO standards of Geometrical Product Specifications recommend the usage of the infinite norm L∞ (min-max) to determine the minimum zone (MZ). This is performed by directly minimizing the peak-to-valley (PV) which is the difference between maximum and minimum deviations of the dataset and the reference surface. Performing data fitting according to L∞ remains a major challenge when considering complex geometries such as aspheres and freeform surfaces.In this work, two algorithms for aspheres minimum zone fitting were implemented and compared on generated reference and measured datasets, namely the Exponential Penalty Function and Primal-Dual Interior Point Method The obtained results show that both methods give accurate values of minimum zone. When the number of points increases, a decay in the latter method’ performances was also noticed especially for calculation time and accuracy of returned minimum zone values.

Beyond the Known: Detecting Novel Feasible Domains Over an Unbounded Design Space

To solve a design problem, sometimes it is necessary to identify the feasible design space. For design spaces with implicit constraints, sampling methods are usually used. These methods typically bound the design space; that is, limit the range of design variables. But bounds that are too small will fail to cover all possible designs, while bounds that are too large will waste sampling budget. This paper tries to solve the problem of efficiently discovering (possibly disconnected) feasible domains in an unbounded design space. We propose a data-driven adaptive sampling technique—ε-margin sampling, which learns the domain boundary of feasible designs and also expands our knowledge on the design space as available budget increases. This technique is data-efficient, in that it makes principled probabilistic trade-offs between refining existing domain boundaries versus expanding the design space. We demonstrate that this method can better identify feasible domains on standard test functions compared to both random and active sampling (via uncertainty sampling). However, a fundamental problem when applying adaptive sampling to real world designs is that designs often have high dimensionality and thus require (in the worst case) exponentially more samples per dimension. We show how coupling design manifolds with ε-margin sampling allows us to actively expand high-dimensional design spaces without incurring this exponential penalty. We demonstrate this on real-world examples of glassware and bottle design, where our method discovers designs that have different appearance and functionality from its initial design set.

Minimax Robust Quickest Change Detection With Exponential Delay Penalties

Quickly detecting changes in the statistical behavior of measurements is important in many applications of control engineering involving fault detection and process monitoring. In this letter, we pose and solve minimax robust Lorden and Bayesian quickest change detection problems for situations where the cost of detection delays compounds exponentially. We show that the detection rules that solve our robust quickest change detection problems are also the rules that solve the standard (non-robust) problems specified by least favorable distributions from uncertainty classes of possible distributions that satisfy a stochastic boundedness condition. In contrast to previous robust quickest change detection results with nonlinear detection delay penalties, our results with exponential delay penalties are exact (i.e., they hold for any false alarm constraint and not only in the asymptotic regime of few false alarms). We illustrate our results through simulations.

A theoretical understanding of self-paced learning

Self-paced learning (SPL) is a recently proposed methodology designed by mimicking through the learning principle of humans/animals. A variety of SPL realization schemes have been designed for different computer vision and pattern recognition tasks, and empirically demonstrated to be effective in these applications. However, the literature is in lack of the theoretical understanding of SPL. Regarding this research gap, this study attempts to provide some new theoretical understanding of the SPL scheme. Specifically, we prove that the solution strategy on SPL accords with a majorization minimization algorithm implemented on an implicit objective function. Furthermore, we found that the loss function contained in this implicit objective has a similar configuration with the non-convex regularized penalty (NCRP) known in statistics and machine learning. Such connection inspires us to discover more intrinsic relationships between the SPL regimes and the NCRP forms, like smoothly clipped absolute deviation (SCAD), logarithmic penalty (LOG) and non-convex exponential penalty (EXP). The insight of the robustness under SPL can then be finely explained. We also analyze the capability of SPL regarding its easy loss-prior-embedding property, and provide an insightful interpretation of the effectiveness mechanism under current SPL variations. Moreover, we design a group-partial-order loss prior, which is especially useful for weakly labeled large-scale data processing tasks. By applying SPL with this loss prior to the FCVID dataset, which is currently one of the largest manually annotated video dataset, our method achieves state-of-the-art performance above existing methods, which further supports the proposed theoretical arguments.

Vector Exponential Penalty Function Method for Nondifferentiable Multiobjective Programming Problems

In this paper, a new vector exponential penalty function method for nondifferentiable multiobjective programming problems with inequality constraints is introduced. First, the case when a sequence of vector penalized optimization problems with vector exponential penalty function constructed for the original multiobjective programming problem is considered, and the convergence of this method is established. Further, the exactness property of a vector exact penalty function method is defined and analyzed in the context of the introduced vector exponential penalty function method. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto solutions of the considered nondifferentiable multiobjective programming problem and the associated vector penalized optimization problem with the vector exact exponential penalty function. This equivalence is established for nondifferentiable vector optimization problems with inequality constraints in which involving functions are r-invex.

Convergence Results of an Augmented Lagrangian Method Using the Exponential Penalty Function

In the present research, an Augmented Lagrangian method with the use of the exponential penalty function for solving inequality constraints problems is considered. Global convergence is proved using the constant positive generator constraint qualification when the subproblem is solved in an approximate form. Since this constraint qualification was defined recently, the present convergence result is new for the Augmented Lagrangian method based on the exponential penalty function. Boundedness of the penalty parameters is proved considering classical conditions. Three illustrative examples are presented.

A homotopy method based on penalty function for nonlinear semidefinite programming

This paper proposes a homotopy method based on a penalty function for solving nonlinear semidefinite programming problems. The penalty function is the composite function of an exponential penalty function, the eigenvalue function and a nonlinear operator mapping. Representations of its first and second order derivatives are given. Using the penalty function, a new homotopy is constructed. Global convergence of a smooth curve determined by the homotopy is proven under mild conditions. In the process of numerically tracing the curve, the method requires just the solution of a linear system of dimension $$n+2$$n+2, whereas a homotopy method proposed by Yang and Yu (Comput Optim Appl 56(1):81---96, 2013) requires a system of dimension $$n+m(m+1)/2+1$$n+m(m+1)/2+1 to be solved, where $$n$$n is the number of variables while $$m$$m is the order of constraint matrix. So, it is expected that the proposed method can improve the efficiency of the method proposed by Yang and Yu. Preliminary numerical experiments are presented and show that the considered algorithm is efficient for some nonlinear semidefinite programming problems.

Convergence of exponential penalty function method for multiobjective fractional programming problems

In this paper, we extend the application of exponential penalty function method for solving multiobjective programming problem introduced by Liu and Feng (2010) to solve multiobjective fractional programming problem and analyze the relationship between weak efficient solutions of penalized problems and multiobjective fractional programming problem. Furthermore, we examine the convergence of this method for multiobjective fractional programming problems.

Multilevel vehicle–dummy design optimization for mass and injury criteria minimization

A vehicle–dummy multidisciplinary design optimization problem is treated as a multilevel system composed of structural and occupant restraint system design elements. The vehicle-based responses and the dummy-based responses are obtained from nonlinear transient dynamic finite element simulations of full frontal impacts and side impacts. The wall thicknesses of a set of energy-absorbing components together with the occupant restraint system control parameters associated with the seat belt and the airbag are treated as design variables and used to optimize the multilevel system to minimize both the structural mass and the selected injury criteria. Each element optimization problem is modeled using the augmented Lagrangian with exponential penalty function formulation. A single-loop coordination strategy is used to solve the multilevel optimization problem. To maximize the computational efficiency, surrogate models are used to approximate the vehicle-based responses and the dummy-based responses. The results of the vehicle–dummy design problem are used to examine the computational efficiency and the accuracy of the decomposed multilevel optimization methodology.

An Adversarial Interpretation of Information-Theoretic Bounded Rationality

Recently, there has been a growing interest in modeling planning with information constraints. Accordingly, an agent maximizes a regularized expected utility known as the free energy, where the regularizer is given by the information divergence from a prior to a posterior policy. While this approach can be justified in various ways, including from statistical mechanics and information theory, it is still unclear how it relates to decision-making against adversarial environments. This connection has previously been suggested in work relating the free energy to risk-sensitive control and to extensive form games. Here, we show that a single-agent free energy optimization is equivalent to a game between the agent and an imaginary adversary. The adversary can, by paying an exponential penalty, generate costs that diminish the decision maker's payoffs. It turns out that the optimal strategy of the adversary consists in choosing costs so as to render the decision maker indifferent among its choices, which is a definining property of a Nash equilibrium, thus tightening the connection between free energy optimization and game theory.

An Exact l 1 Exponential Penalty Function Method for Multiobjective Optimization Problems with Exponential-Type Invexity

The purpose of this paper is to devise exact l1 exponential penalty function method to solve multiobjective optimization problems with exponential-type invexity. The conditions governing the equivalence of the (weak) efficient solutions to the vector optimization problem and the (weak) efficient solutions to associated unconstrained exponential penalized multiobjective optimization problem are studied. Examples are given to illustrate the obtained results.

On the martingale and free-boundary approaches in sequential detection problems with exponential penalty for delay

We study the connection between the martingale and free-boundary approaches in sequential detection problems for the drift of a Brownian motion, under the assumption of exponential penalty for the delay. By means of the solution of a suitable free-boundary problem, we show that the reward process can be decomposed into the product between a gain function of the boundary point and a positive martingale inside the continuation region.

Chebyshev fitting of complex surfaces for precision metrology

The form qualities of precision components are essential for their functionalities. The Peak-to-Valley parameters are widely adopted to assess the form accuracies of optical components. The commonly used least squares method is prone to over-estimation, thus the Chebyshev fitting should in turn be implemented. In this paper the original minimax optimisation problem is converted into an unconstrained differentiable minimisation problem by the exponential penalty functions. The fitting accuracy and numerical stability are balanced by employing an active-set strategy and adjusting the configuration parameters adaptively. Finally some benchmark data sets are applied to demonstrate the validity and efficiency of this method.