First-order Conditions Research Articles

Anion- and water-facilitated oxidative carbon–carbon bond cleavage and diketonate carboxylation in Cu(II) chlorodiketonate complexes

The O2-dependent carbon–carbon (CC) bond cleavage reactions of the mononuclear Cu(II) chlorodiketonate complexes [(6-Ph2TPA)Cu(PhC(O)CClC(O)Ph)]ClO4 (1-ClO4) and [(bpy)Cu(PhC(O)CClC(O)Ph)(ClO4)] (3-ClO4) have been further examined in terms of their anion and water dependence. The bpy-ligated Cu(II) chlorodiketonate complex 3-ClO4 is inherently more reactive with O2 than the 6-Ph2TPA-ligated analog 1-ClO4. Added chloride is needed to facilitate O2 reactivity for 1-ClO4 but not for 3-ClO4 at 25(1) °C. Evaluation of kobs for the reaction of 1-ClO4 with O2 under pseudo first-order conditions as a function of the amount of added chloride ion produced saturation type behavior. The bpy-ligated 3-ClO4 exhibits different behavior, with rate enhancement resulting from both the addition of chloride ion and water. Computational studies indicate that the presence of water lowers the barrier for O2 activation for 3-ClO4 by ∼12 kcal/mol whereas changing the anion from perchlorate to chloride has a smaller effect (lowering of the barrier by ∼3 kcal/mol). Notably, the effect of water for 3-ClO4 is of similar magnitude to the barrier-lowering chloride effect found in the O2 activation pathway for 1-ClO4. Thus, both systems involve lower energy O2 activation pathways available, albeit resulting from different ligand effects. Probing the effect of added benzoate anion, it was found that the chloro substituent in the diketonate moiety of 1-ClO4 and 3-ClO4 will undergo displacement upon treatment of each complex with tetrabutyl ammonium benzoate to give Cu(II) benzoyloxydiketonate complexes (4 and 5). Complexes 4 and 5 exhibit slow O2-dependent CC cleavage in the presence of added chloride ion. These results are discussed in the context of the chemistry identified for various divalent metal chlorodiketonate complexes, which have relevance to catalytic systems and metalloenzymes that mediate O2-dependent CC cleavage within diketonate substrates.

Acute ethanol disrupts conditioned inhibition in the male rat.

Alcohol can disrupt conditioned sexual inhibition (CSI) established by first-order conditioning in male rats. CSI can also be induced using second-order conditioning, during which male rats are trained to associate a neutral odor with a nonreceptive female. As a result, when given access to two receptive females (one scented and one unscented) during a copulatory preference test, they display CSI toward the scented female. The present study examined the effect of low-to-moderate doses of alcohol on CSI and brain activation following exposure to alcohol and the olfactory cue alone. Sexually-naïve Long-Evans rats received alternate conditioning sessions with unscented receptive or scented (almond extract) non-receptive females. Following the conditioning phase, males were injected with saline, alcohol 0.5g/kg or 1g/kg, 45min before a copulatory test with two receptive females, with one bearing the olfactory cue. Fos activation was later assessed, following exposure to alcohol and the olfactory cue alone, in several brain regions involved in the expression and regulation of male sexual behavior. While males in the saline group displayed sexual avoidance towards the scented female, those injected with alcohol before the copulatory test, regardless of the dose, copulated indiscriminately with both females. Subsequent exposure to alcohol and the olfactory cue alone induced different Fos expression between groups in several brain regions. Low to moderate doses of alcohol disrupt conditioned sexual inhibition in male rats and induce a differential pattern of neural activation, particularly in regions involved in the expression and regulation of sexual behavior.

Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system’s state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed.

Spectral-like conjugate gradient methods with sufficient descent property for vector optimization.

Several conjugate gradient (CG) parameters resulted in promising methods for optimization problems. However, it turns out that some of these parameters, for example, 'PRP,' 'HS,' and 'DL,' do not guarantee sufficient descent of the search direction. In this work, we introduce new spectral-like CG methods that achieve sufficient descent property independently of any line search (LSE) and for arbitrary nonnegative CG parameters. We establish the global convergence of these methods for four different parameters using Wolfe LSE. Our algorithm achieves this without regular restart and assumption of convexity regarding the objective functions. The sequences generated by our algorithm identify points that satisfy the first-order necessary condition for Pareto optimality. We conduct computational experiments to showcase the implementation and effectiveness of the proposed methods. The proposed spectral-like methods, namely nonnegative SPRP, SHZ, SDL, and SHS, exhibit superior performance based on their arrangement, outperforming HZ and SP methods in terms of the number of iterations, function evaluations, and gradient evaluations.

Parameter identification of a reaction-diffusion predator-prey system based on optimal control theory

This paper takes a reaction-diffusion predator-prey system with ratio-dependent Holling III functional response function and Leslie-Gower term into consideration. First of all, the system model is proposed on the basis of basic biological assumptions and previous work, and the existence conditions of the equilibrium point of the system are discussed. Secondly, under the assumption of the existence of equilibrium point, the Turing instability necessary conditions induced by diffusion are investigated. Thirdly, optimal control theory is derived, and the adjoint system and the first-order optimization condition are established. Fourthly, parameter identification based on optimal control is studied, and the technique is extended to network structure. Finally, extensive numerical simulations, including Turing pattern, Normalized Population High Distribution Area (NPHDA) diagram and parameter identification, are carried out to illustrate and validate the analytical results. For the system dynamics phenomena, the results from different perspectives effectively demonstrate that the theoretical findings, numerical simulations and natural reality are identical. In terms of the parameter identification of continuous model and network model, the efficiency and accuracy of various algorithms are fully tested.

Implications of Context Effects in Consumer Utility Models for Optimal Product Design and Differentiation

Abstract Consumer choice models used in optimal product design typically ignore potential context effects by assuming the utility of each product is independent of the attributes of other products in the choice set. We characterize implications of context effects for profit-optimal designs by deriving the first-order conditions of the design problem under alternative utility formulations, and we propose a utility function that incorporates context effects and has well-defined optimal design solutions for all products in the choice set. We then conduct a discrete choice survey experiment of automobile options and find statistically significant context-effect parameters and superior out-of-sample prediction when context-effect parameters are used in both logit and mixed logit models. These results suggest that context effects can be important in engineering design contexts and have the potential to affect optimal design differentiation.

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

Turing instability analysis and parameter identification based on optimal control and statistics method for a rumor propagation system.

This study establishes a reaction-diffusion system to capture the dynamics of rumor propagation, considering two possibilities of contact transmission. The sufficient and necessary conditions for a positive equilibrium point are provided, and the Turing instability conditions for this equilibrium point are derived. Furthermore, utilizing variational inequalities, a first-order necessary condition for parameter identification based on optimal control is established. During the numerical simulation process, the correctness of the Turing instability conditions is verified, and optimal control-based parameter identification is applied to the target pattern. Additionally, statistical methods are employed for pattern parameter identification. The identification results demonstrate that optimal control-based parameter identification exhibits higher efficiency and accuracy. Finally, both theories' parameter identification principles are extended to a small-world network, yielding consistent conclusions with continuous space.

WITHDRAWN: Kinetics and mechanism of the reactions of alkanediamine-linked dinuclear platinum(II) complexes with some biorelevant-nucleophiles

The kinetics and mechanism of substitition reactions of dinuclear platinum(II) complexes of the general formula [{Pt(H2O)}2(N,N,N′,N′-tetrakis(2-pyridylmethyl)-N(CH2)n-N](ClO4)4,viz., n = 2 (PtEn); 3 (PtProp); 6 (PtHex); 8 (PtOct) and 10 (PtDec) were investigated in 0.10 M aqueous perchloric acid medium as a function of concentration of l-glutathione (Glu) and dl-penicilliamine (Pen) nucleophiles and temperature under pseudo first-order conditions using UV–visible spectrophotometry. The reactivity data of the dinuclear Pt(II) complexes were compared with those of their mononuclear analogues of the form [Pt(H2O)(N,N-bis(2-pyridylmethyl)-N(CH2)n-CH3](ClO4)2, n = 1 (Ptbpea); 2 (Ptbppa); 5 (Ptbpha) and 9 (Ptbpda) under the same experimental conditions. The observed rate constants for the substitution of the aqua ligands follow the simple rate law: kobs = k2[Nu]. The reactivity of the complexes increases as the chain length of the linker or tail increases except Prop which showed a higher reactivity towards the nucleophiles than the rest of the complexes. The overall trend of reactivity for the dinuclear platinum(II) complexes is PtEn < PtHex < PtOct < PtDec < PtProp. The reactivity of the mononuclear complexes follows the same trend as the dinuclears and increases in the order Ptbpea < Ptbppa < Ptbpha < Ptbpda. In all cases, the mononuclear complexes are more reactive than their dinuclear analogues, which is an indication of the steric retardation on the rate of substitution due to the conformations of the dinuclear platinum(II) complexes. Glu was found to be a better nucleophile than Pen. The relatively small and positive values for the enthalpy of activation, negative values for entropy of activation and the dependence of the second-order rate constants on the concentration of entering nucleophiles supports an associatively activated mechanism.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

Space-Time Least-Squares Finite Element Methods for Parabolic Distributed Optimal Control Problems

Abstract We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier. Our final formulation fulfills the Babuška–Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem. Numerical experiments confirm our theoretical findings.

An approach to nonlinear optimisation via set stabilisation of dynamical systems with an application to synchronous machines

An approach to constraint nonlinear optimisation is proposed, where the variables of the objective function and of the constraint function belong to the state of a nonlinear dynamical system. A nonlinear closed-loop feedback system is designed, whose solutions converge to a target set consisting of the optimal points of the underlying optimisation problem. The feedback law for the control input of the nonlinear dynamical system is designed by exact linearisation of a dynamical formulation of a first-order optimality condition. For the problem formulation and the stability analysis of the nonlinear closed-loop feedback system, the notion of V-detectability is considered. An application to energy-optimal feed-forward torque control for different types of synchronous machines is presented, which demonstrates the usefulness of the method. In this regard, anisotropic permanent magnet synchronous machines and electrically excited synchronous machines are covered.

Equivalent analysis model of a GaN LED used as a receiver

In general, visible light communication (VLC) uses LEDs as transmitters. However, LEDs can serve as receivers to construct a simple duplex VLC system that uses only two LEDs instead of one LED and one photo-diode (PD). There is a lack of effective equivalent analysis models for characterizing and evaluating the inherent behavioral characteristics of LEDs used as receivers. This paper presents an equivalent analysis model for GaN LEDs as receivers. First, based on the proposed receiving equivalent circuit model, a third-order signal transmission mathematical analysis model is established, revealing the transmission relationship between the photocurrent and output voltage. Further research is conducted on the impact of parameter changes on the bandwidth, and the model can be simplified into a first-order low-pass mathematical analysis model under specific conditions, providing theoretical support for improving the bandwidth of LED receiving applications. The experimental results also confirm the theoretical predictions. This research result holds significant importance for revealing the intrinsic mechanisms and the improved optical communication performance of LEDs for effective reception.

Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions

Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h2 +τ). The primary research method employed is the method of energy inequalities.Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.

Solutions for Poissonian stopping problems of linear diffusions via extremal processes

We develop a general yet simple technique for solving Poissonian timing problems of linear diffusions by relying on the close connection of the extremal processes and the first passage times of the underlying diffusion. We provide a closed-form representation of the expected value gained by employing an ordinary first passage time-based stopping strategy. This approach simplifies the determination of the optimal policy, transforming it into an analysis of ordinary first-order optimality conditions. We relate our findings to various existing approaches for solving stopping problems of linear diffusions and express the optimality conditions in a single boundary setting in a form familiar from optimal stopping of Lévy-processes.

Trust Region Methods for Nonconvex Stochastic Optimization beyond Lipschitz Smoothness

In many important machine learning applications, the standard assumption of having a globally Lipschitz continuous gradient may fail to hold. This paper delves into a more general (L0, L1)-smoothness setting, which gains particular significance within the realms of deep neural networks and distributionally robust optimization (DRO). We demonstrate the significant advantage of trust region methods for stochastic nonconvex optimization under such generalized smoothness assumption. We show that first-order trust region methods can recover the normalized and clipped stochastic gradient as special cases and then provide a unified analysis to show their convergence to first-order stationary conditions. Motivated by the important application of DRO, we propose a generalized high-order smoothness condition, under which second-order trust region methods can achieve a complexity of O(epsilon(-3.5)) for convergence to second-order stationary points. By incorporating variance reduction, the second-order trust region method obtains an even better complexity of O(epsilon(-3)), matching the optimal bound for standard smooth optimization. To our best knowledge, this is the first work to show convergence beyond the first-order stationary condition for generalized smooth optimization. Preliminary experiments show that our proposed algorithms perform favorably compared with existing methods.

An adaptive phase-field method for structural topology optimization

In this work, we develop an adaptive algorithm for the efficient numerical solution of the minimum compliance problem in topology optimization. The algorithm employs the phase field approximation and continuous density field. The adaptive procedure is driven by two residual type a posteriori error estimators, one for the state variable and the other for the first-order optimality condition of the objective functional. The adaptive algorithm is provably convergent in the sense that the sequence of numerical approximations generated by the adaptive algorithm contains a subsequence convergent to a solution of the continuous first-order optimality system. We provide several numerical simulations to show the distinct features of the algorithm.

On generalized Nash equilibrium problems in infinite-dimensional spaces using Nikaido–Isoda type functionals

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE.

Viscous dissipation effects on slip flow heat transfer in rhombic microchannels

This study aimed to numerically investigate the viscous dissipation effect on forced convection in rhombic microchannels for gases in a slip flow regime. The numerical analysis was carried out by assuming a 2D steady-state flow. The solution of governing equations was obtained by adopting the finite element method and assuming that the fluid is Newtonian, with constant thermophysical properties, and in a fully developed laminar flow regime. The solution of the momentum equation is obtained by considering a first-order boundary condition, while the solution of the energy balance equation is obtained by assuming a constant wall heat flux (H2 boundary condition) and taking into account the wall temperature jump. The validation of the numerical model was carried out using the data available in the scientific literature. The numerical outcomes obtained for several values of the acute angle of the rhombus, the Knudsen number (i.e., rarefaction effects), and the Brinkman number (i.e., viscous dissipation effects) reveal that viscous forces play an opposite role with respect to rarefaction and significantly affect the convective heat transfer coefficient, especially for low Knudsen numbers and for high values of the acute angle. In particular, it was observed that Nu is significantly affected by the Brinkman number for acute angles higher than 50°.

Optimal Sizing of Hybrid Renewable Energy System Using Two-Stage Stochastic Programming

Stochastic programming has become increasingly vital in energy applications, especially in the context of the growing need for renewable energy solutions. This paper presents a significant advancement in this field by introducing an efficient and robust algorithm for optimally sizing hybrid renewable energy systems. Utilizing a two-stage stochastic programming approach, the proposed algorithm addresses the challenges posed by the unpredictability of renewable energy sources. The proposed solution leverages the three-block alternating direction method of multipliers (ADMM), a cutting-edge technique that facilitates parallel computation and enhances computational efficiency. The distinctiveness of this method lies in its ability to solve complex stochastic optimization problems without compromising the mathematical integrity of the model. This is achieved by applying first-order optimality conditions, ensuring both robustness and efficacy. To demonstrate the practical applicability and superiority of the algorithm, a case study was conducted in a rural area of South Africa. The proposed algorithm was applied to design an optimal hybrid renewable energy system, and its performance was compared against traditional methods such as progressive hedging and Monte Carlo techniques. Results affirm the superiority of the approach, saving approximately 8.16% capital cost when compared to progressive hedging. In addition, the proposed algorithm outperforms the Monte Carlo method both in terms of CPU time and the number of cost function evaluations.