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In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. We then extend the

The joint action-value function (JAVF) plays a key role in the centralized training of multi-agent deep reinforcement learning (MADRL)-based algorithms using the value function decomposition (VFD) and in the generating process of a collaborative policy between agents. However, under the influence of multiple factors such as environmental noise, inadequate exploration and iterative updating mechanism, estimation bias is inevitably introduced, causing its overestimation problem, which in turn prevents agents from obtaining accurate reward signals during the learning process, and fails to correctly approximate the optimal policy. To address this problem, this paper first analyzes the causes of joint action-value function overestimation, gives the corresponding mathematical proofs and theoretical derivations, and obtains the lower bound of the overestimation error; then, a MADRL overestimation reduction method based on the multi-step weighted double estimation named λWD QMIX is proposed. Specifically, the λWD QMIX method effectively achieves more stable and accurate JAVF estimation results using the bias correction estimation mechanisms based on the weighted double estimation and multi-step updating based on eligibility trace backup, without additionally adding or changing any network structure. The results of a series of experiments on the StarCraft II micromanipulation benchmark show that the proposed λWD QMIX algorithm can effectively improve the final performance and learning efficiency of the baseline algorithm, and can be seamlessly integrated with the partially MADRL algorithms based on communication learning.

In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given.

This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization called the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do not---to a large extent---differ from those known for the conventional problem. It has already been well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This is related to the local Lipschitz continuity of the optimal value function of an optimization problem constrained by solutions to another optimization problem; this function is labeled here as the two-level value function. More generally, we conduct a detailed sensitivity analysis for value functions of mathematical programs with extended complementarity constraints. The results obtained in this vein are applied to the two-level value function and then to the original optimistic formulation of the bilevel optimization problem, for which we derive verifiable stationarity conditions of various types entirely in terms of the initial data. [PUBLICATION ABSTRACT]

We establish some elementary results on solutions to the Bellman equation without introducing any topological assumption. Under a small number of conditions, we show that the Bellman equation has a unique solution in a certain set, that this solution is the value function, and that the value function can be computed by value iteration with an appropriate initial condition. In addition, we show that the value function can be computed by the same procedure under alternative conditions. We apply our results to two optimal growth models: one with a discontinuous production function and the other with \"roughly increasing\" returns.

In this article, we study the uniqueness of differential polynomials P(f)=f_1^p P(f_1) and P[f] generated by meromorphic functions f and g respectively sharing a small function. Our results generalises the result due to Harina P. Waghamore and Husna V. [7].

In this paper, we introduce the concept of a representative value function in a group decision context. We extend recently proposed methods UTA GMS -GROUP and UTADIS GMS -GROUP with selection of a compromise and collective preference model which aggregates preferences of several decision makers (DMs) and represents all instances of preference models compatible with preference information elicited from DMs. The representative value function is built on results of robust ordinal regression, so its representativeness can be interpreted in terms of robustness concern. We propose a few procedures designed for multiple criteria ranking, choice, and sorting problems. The use of these procedures is conditioned by both satisfying different degrees of consistency of the preference information provided by all DMs, as well as by some properties of particular decision making situations. The representative value function is intended to help the DMs to understand the robust results, and to provide them with a compromise result in case of conflict between the DMs.

In this paper, we consider a mean-field stochastic differential equation, also called the McKean-Vlasov equation, with initial data (t, x) ∈ [0, T] × ℝd, whose coefficients depend on both the solution $X_s^{t,x}$ and its law. By considering square integrable random variables ξ as initial condition for this equation, we can easily show the flow property of the solution $X_s^{t,\\xi }$ of this new equation. Associating it with a process $X_s^{t,x,P\\xi }$ which coincides with $X_s^{t,\\xi }$, when one substitutes ξ for x, but which has the advantage to depend on ξ only through its law Pξ, we characterize the function $V\\left( {t,x,P\\xi } \\right) = E\\left[ {\\Phi \\left( {X_T^{t,x,P\\xi },{P_{X_T^{l,\\xi }}} \\right)} \\right]$ under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of mean-field type, involving the first- and the second-order derivatives of V with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first- and second-order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Itô formula. In our approach, we use the notion of derivative with respect to a probability measure with finite second moment, introduced by Lions in [Cours au Collège de France: Théorie des jeu à champs moyens (2013)], and we extend it in a direct way to the second-order derivatives.

Landscape ecology is in a position to become the scientific basis for sustainable landscape development. When spatial planning policy is decentralised, local actors need to collaborate to decide on the changes that have to be made in the landscape to better accommodate their perceptions of value. This paper addresses two prerequisites that landscape ecological science has to meet for it to be effective in producing appropriate knowledge for such bottom-up landscape-development processes--it must include a valuation component, and it must be suitable for use in collaborative decision-making on a local scale. We argue that landscape ecological research needs to focus more on these issues and propose the concept of landscape services as a unifying common ground where scientists from various disciplines are encouraged to cooperate in producing a common knowledge base that can be integrated into multifunctional, actor-led landscape development. We elaborate this concept into a knowledge framework, the structure-function-value chain, and expand the current pattern-process paradigm in landscape ecology with value in this way. Subsequently, we analyse how the framework could be applied and facilitate interdisciplinary research that is applicable in transdisciplinary landscape-development processes.

Background Drought is a major limiting factor seriously influencing worldwide soybean production and its impact on yield, morphological and physiological traits depend on the timing it occurs and the intensity of water shortage. Only limited research has however been conducted on identifying the drought-tolerant genotypes at different growth stages (vegetative growth phase, reproductive growth phase and the whole growth phase) as well as evaluate the effectiveness and reliability of multiple phenotypic and yield-related characteristics in soybean. Results Two pot experiments and a 2-year field experiment were conducted to evaluate soybean drought tolerance at different growth stages. The membership function value of drought tolerance (MFVD) was used to identify drought-resistant cultivars during vegetative growth phase and reproductive growth stage; the relative drought index (RDI) of yield was used to assess drought-resistant cultivars during the whole growing period. In this study, regression models built based on MFVD indicated that the variation of drought tolerant coefficient (DC) of R/S, TRL, LAI and RSR could explain 73.70% of the total variation at vegetative growth phase. However, higher heritability only found in LAI and RSR, indicating the two traits could serve as reliable criteria for drought evaluation. Similarly, the DC of SPP, YPP, PH, PB, MSNN and STB could explain 94.30% of the total variation in MFVD according to stepwise multiple linear regression analyses at reproductive growth phase. Thus, these six traits were identified as indicators for screening drought resistance genotypes in soybean. In addition, correlation analysis revealed that the MFVD was significantly positively correlated with the DC RB , DC R/S , DC RSA , DC RSR and DC RBR at vegetative growth phase and DC YPP , DC SPP , DC RB, and DC PB at reproductive growth phase. This indicated that these traits were closely related to the drought resistance of plants. Conclusions LD24, JD36 and TF31 of vegetative growth phase, and TD37 and LD26 of reproductive growth phase were identified with drought tolerant and highly drought tolerant, respectively. Moreover, 30 accessions with drought tolerance were screened in the field trial and could be applied for the drought resistance of other genotypes by cross-breeding.