Sufficient Efficiency Conditions Research Articles

Efficient nonparametric estimation of generalised autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalised autocovariance function of a stationary random process. The generalised autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. We consider two cases: the fixed bandwidth design and the adaptive bandwidth design. The general result on the asymptotic efficiency, established for linear processes, is then applied to the class of stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule–Walker system of equations in the generalised autocovariance estimator.

Asymptotic Relative Efficiency of the Kendall and Spearman Correlation Statistics

A necessary and sufficient condition for Pitman's asymptotic relative efficiency of the Kendall and Spearman correlation statistics for the independence test to be $1$ is given, in terms of certain smoothness and nondegeneracy properties of the model. Corresponding easy-to-use and broadly applicable sufficient conditions are obtained. These conditions hold for most well-known models of dependence.

Robust Efficiency Conditions in Multiple-Objective Fractional Variational Control Problems

The aim of this study is to investigate multi-dimensional vector variational problems considering data uncertainty in each of the objective functional and constraints. We establish the robust necessary and sufficient efficiency conditions such that any robust feasible solution could be the robust weakly efficient solution for the problems under consideration. Emphatically, we present robust efficiency conditions for multi-dimensional scalar, vector, and vector fractional variational problems by using the notion of a convex functional.

Efficient Nonparametric Estimation of Generalized Autocovariances

This paper provides a necessary and sufficient condition for asymptotic efficiency of a nonparametric estimator of the generalized autocovariance function of a stationary random process. The generalized autocovariance function is the inverse Fourier transform of a power transformation of the spectral density and encompasses the traditional and inverse autocovariance functions as particular cases. A nonparametric estimator is based on the inverse discrete Fourier transform of the power transformation of the pooled periodogram. The general result on the asymptotic efficiency is then applied to the class of Gaussian stationary ARMA processes and its implications are discussed. Finally, we illustrate that for a class of contrast functionals and spectral densities, the minimum contrast estimator of the spectral density satisfies a Yule-Walker system of equations in the generalized autocovariance estimator.

Second-Order Necessary and Sufficient Optimality Conditions for Constrained Vector Equilibrium Problem with Applications

In this paper, we study a generalized convex vector equilibrium problem with cone and set constraints in real Banach spaces. We provide some basic characterizations on generalized convexity for the first- and second-order directional derivatives. We obtain Kuhn–Tucker second-order necessary and sufficient optimality conditions for efficiency to such problem under suitable assumptions on the generalized convexity of objective and constraint functions. As an application, we present Kuhn–Tucker second-order necessary and sufficient optimality conditions to a generalized convex vector variational inequality problem and a generalized convex vector optimization problem with constraints. Some examples are also given to demonstrate the main results of the paper.

Strong Karush–Kuhn–Tucker optimality conditions for weak efficiency in constrained multiobjective programming problems in terms of mordukhovich subdifferentials

Based on the notation of Mordukhovich subdifferentials (Mordukhovich in Variational analysis and generalized differentiation I: basic theory, Springer, Berlin, 2006; Variational analysis and generalized differentiation II: applications, Springer, Berlin, 2006; Variational analysis and applications, Springer, Berlin, 2018), we establish strong Karush–Kuhn–Tucker type necessary optimality conditions for the weak efficiency of a nonsmooth nonconvex multiobjective programming problem with set, inequality and equality constraints. We also provide several new definitions for the Mordukhovich-pseudoconvexity and Mordukhovich-quasiconvexity with extended-real-valued functions, and then provide sufficient optimality conditions for weak efficiency to such problem in terms of Mordukhovich subdifferentials.

Process Control and Energy Efficiency.

We review the impact of control systems and strategies on the energy efficiency of chemical processes. We show that, in many ways, good control performance is a necessary but not sufficient condition for energy efficiency. The direct effect of process control on energy efficiency is manyfold: Reducing output variability allows for operating chemical plants closer to their limits, where the energy/economic optima typically lie. Further, good control enables novel, transient operating strategies, such as conversion smoothing and demand response. Indirectly, control systems are key to the implementation and operation of more energy-efficient plant designs, as dictated by the process integration and intensification paradigms. These conclusions are supported with references to numerous examples from the literature.

Strict efficiency of a multi-product supply-demand network equilibrium model

In this paper, we consider a kind of proper efficiency, namely strict efficiency, of a multi-product supply-demand network equilibrium model. We prove that strict equilibrium pattern flows with both a single criterion and multiple criteria are equivalent to vector variational inequalities. In the case of multiple criteria, we provide necessary and sufficient conditions for strict efficiency in terms of vector variational inequalities by using Gerstewitz's function without any convexity assumptions.

Characterizations of Benson proper efficiency of set-valued optimization in real linear spaces

Abstract A new class of generalized convex set-valued maps termed relatively solid generalized cone-subconvexlike maps is introduced in real linear spaces not equipped with any topology. This class is a generalization of generalized cone-subconvexlike maps and relatively solid cone-subconvexlike maps. Necessary and sufficient conditions for Benson proper efficiency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality. The results generalize and improve some corresponding ones in the literature. Some examples are afforded to illustrate our results.

Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming

ABSTRACTSecond-order optimality conditions for the vector nonlinear programming problems with inequality constraints and continuously differentiable data are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce notions of a second-order pseudoconvex vector function and KKT-pseudoconvex vector problem with inequality constraints. We derive necessary and sufficient conditions for efficiency in the vector problem with inequality constraints. Three examples are presented.

Efficiency Advantages and Incentive Mechanism of PPPs: A Qualitative Comparative Analysis under the Chinese Scenario

Based on the PPP efficiency system which consists of allocation efficiency, process efficiency, and individual efficiency, we use qualitative comparison analysis of fuzzy sets to study the efficiency advantages of the public-private partnership under the Chinese scenario. The findings are as follows: (1) like public-private partnerships, Chinese-style PPPs have also failed to achieve cooperation. (2) High allocation efficiency can be achieved if competition in bidding processes can be ensured; when bidding procedures cannot be guaranteed to compete, alternatives to high allocation efficiency are either privatized or allocated directly to enterprises that can enable economies of scale; individual effort is a source of allocation efficiency. (3) Competition and economies of scale are necessary conditions for high process efficiency. The private sector’s ownership of assets is a sufficient condition for high process efficiency. (4) High individual efficiency can be achieved if individual efforts can be ensured, and high individual efficiency can also be achieved by the competition of bidding procedures or economies of scale when it is impossible to ensure high levels of individual effort. Privatization is the perfect incentive for high individual efficiency when the competition in the bidding process, individual efforts, and economies of scale cannot be guaranteed.

On a Sufficient Condition for Weak Sharp Efficiency in Multiobjective Optimization

In this paper, we provide sufficient conditions entailing the existence of weak sharp efficient points of a multiobjective optimization problem. The approach uses variational analysis techniques, like regularity and subregularity of the diagonal subdifferential map related to a suitable scalar equilibrium problem naturally associated to the multiobjective optimization problem.

Efficiency conditions in vector control problems governed by multiple integrals

In this paper, we formulate and prove necessary and sufficient optimality conditions in multiobjective control problems which involve multiple integrals. Under $$ \displaystyle (\rho , b) $$ -quasiinvexity assumptions, sufficient efficiency conditions for a feasible solution are derived.

New efficiency conditions for multiobjective interval-valued programming problems

In this paper, we focus on necessary and sufficient efficiency conditions for optimization problems with multiple objectives and a feasible set defined by interval-valued functions. A new concept of Fritz-John and Karush–Kuhn–Tucker-type points is introduced for this mathematical programming problem based on the gH-derivative concept. The innovation and importance of these concepts are presented from a practical and computational point of view. The problem is approached directly, without transforming it into a real-valued programming problem, thereby attaining theoretical results that are more powerful and computationally more efficient under weaker hypotheses. We also provide necessary conditions for efficiency, which have been inexistent in the relevant literature to date. The identification of necessary conditions is important for the development of future computational optimization techniques in an interval-valued environment. We introduce new generalized convexity notions for gH-differentiable interval-valued problems which are a generalization of previous concepts and we prove a sufficient efficiency condition based on these concepts. Finally, the efficiency conditions for deterministic programming problems are shown to be particular instances of the results proved in this paper. The theoretical developments are illustrated and justified through several numerical examples.

Sufficient condition for partial efficiency in a bicriteria nonlinear cutting stock problem

This work presents a sufficient criteria for partial efficient solutions of the cutting stock problem with two objectives. We consider two important objectives for an industry: number of processed objects (cost of raw materials) and number of different patterns (cost of setup). These optimality results are established through a new approach based on connections between discrete optimization and continuous vector optimization.

Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization

We provide necessary and sufficient conditions for robust efficiency (in the sense of Ehrgott et al., 2014) to multiobjective optimization problems that depend on uncertain parameters. These conditions state that a solution is robust efficient (under minimization) if it is optimal to a strongly increasing scalarizing function, and only if it is optimal to a strictly increasing scalarizing function. By counterexample, we show that the necessary condition cannot be strengthened to convex scalarizing functions, even for convex problems. We therefore define and characterize a subset of the robust efficient solutions for which an analogous necessary condition holds with respect to convex scalarizing functions. This result parallels the deterministic case where optimality to a convex and strictly increasing scalarizing function constitutes a necessary condition for efficiency. By a numerical example from the field of radiation therapy treatment plan optimization, we illustrate that the curvature of the scalarizing function influences the conservatism of an optimized solution in the uncertain case.

On gap functions for nonsmooth multiobjective optimization problems

A set-valued gap function, \(\phi \), existing in the literature for smooth and nonsmooth multiobjective optimization problems is dealt with. It is known that \(0\in \phi (x^*)\) is a sufficient condition for efficiency of a feasible solution \(x^*\), while the converse does not hold. In the current work, the converse of this assertion is proved for properly efficient solutions. Afterwards, to avoid the complexities of set-valued maps some new single-valued gap functions, for nonsmooth multiobjective optimization problems with locally Lipschitz data are introduced. Important properties of the new gap functions are established.

Measure-Transformed Quasi-Maximum Likelihood Estimation

In this paper, the Gaussian quasi-maximum likelihood estimator (GQMLE) is generalized by applying a transform to the probability distribution of the data. The proposed estimator, called measure-transformed GQMLE (MT-GQMLE), minimizes the empirical Kullback–Leibler divergence between a transformed probability distribution of the data and a hypothesized Gaussian probability measure. By judicious choice of the transform we show that, unlike the GQMLE, the proposed estimator can gain sensitivity to higher order statistical moments and resilience to outliers leading to significant mitigation of the model mismatch effect on the estimates. Under some mild regularity conditions, we show that the MT-GQMLE is consistent, asymptotically normal and unbiased. Furthermore, we derive a necessary and sufficient condition for asymptotic efficiency. A data driven procedure for optimal selection of the measure transformation parameters is developed that minimizes the trace of an empirical estimate of the asymptotic mean-squared-error matrix. The MT-GQMLE is applied to linear regression and source localization and numerical comparisons illustrate its robustness and resilience to outliers.

Role of exponential type random invexities for asymptotically sufficient efficiency conditions in semi-infinite multi-objective fractional programming.

First a new notion of the random exponential Hanson–Antczak type (alpha ,beta ,gamma ,xi ,eta ,rho ,h(cdot ,cdot ,cdot ),theta )-V-invexity is introduced, which generalizes most of the existing notions in the literature, second a random function h(cdot ,cdot ,cdot ) of the second order is defined, and finally a class of asymptotically sufficient efficiency conditions in semi-infinite multi-objective fractional programming is established. Furthermore, several sets of asymptotic sufficiency results in which various generalized exponential type HA(alpha ,beta ,gamma ,xi ,eta ,rho ,h(cdot ,cdot ,cdot ),theta )-V-invexity assumptions are imposed on certain vector functions whose components are the individual as well as some combinations of the problem functions are examined and proved. To the best of our knowledge, all the established results on the semi-infinite aspects of the multi-objective fractional programming are new, which is a significantly new emerging field of the interdisciplinary research in nature. We also observed that the investigated results can be modified and applied to several special classes of nonlinear programming problems.

Uncertainty Regarding Interpretation of the “Negligence Rule” and Its Implications for the Efficiency of Outcomes

Abstract There are two ways that the negligence rule is interpreted. Under one interpretation a negligent injurer is liable for the entire harm to the victim; and under the other interpretation a negligent injurer is liable only for that part of the harm which can be ascribed to his negligence. Both these versions are efficient. However, if there is uncertainty regarding whether the court will be employing the full liability version or the incremental liability version for determining the liability of a negligent injurer, notwithstanding the fact that both the versions are efficient, inefficiency is possible. It is shown in the paper that a necessary and sufficient condition for efficiency in all cases is that the subjective probability with which the injurer expects the standard version to be employed must be greater than or equal to the subjective probability with which the victim expects the standard version to be employed. For the subset of applications without complementarities in the cares of the two parties and which are such that the total social costs are minimized at a unique care-configuration, it is shown that efficiency obtains regardless of the subjective probabilities with which the parties expect the two versions. One very important conclusion that emerges from the analysis of this paper is that when courts employ more than one liability rule, even if all the employed rules are efficient, the efficiency of all outcomes cannot be taken for granted merely on the ground of the efficiency of the employed rules.