Minimax Research Articles

Another look at Schrödinger equations with prescribed mass

In this paper, we investigate the existence of solutions for the nonlinear Schrödinger equation −Δu−λu=f(u) in RN with an L2-constraint in the Sobolev subcritical case when f possesses several weaker L2-supercritical conditions and in the Sobolev critical case when f(u)=μ|u|q−2u+|u|2⁎−2u with μ>0 and 2<q<2⁎=2NN−2 allowing to be L2-subcritical, L2-critical or L2-supercritical, where N≥3 is the dimension, λ∈R and f∈C(R,R). By establishing several new critical point theorems on a manifold, we introduce a new variational approach which enables us to weaken the previous L2-supercritical conditions in the Sobolev subcritical case and also to present an alternative scheme for all 2<q<2⁎, which is technically simpler compared to the Ghoussoub minimax principle [7] involving topological arguments, to construct bounded (PS) sequences on a manifold when the reaction performs as a mixed dispersion. In particular, the analysis developed in this paper also allows to control the energy level in the Sobolev critical case in a unified way whether N=3 or N≥4. Furthermore, this new approach can provide a new look at what were expected by Soave [13] and by Jeanjean-Le [10] and may it can be applied and modified to more related variational problems.

Minimax solutions of Hamilton-Jacobi equations in dynamic optimization problems for hereditary systems

A survey of the results concerning the development of the theory of Hamilton-Jacobi equations for hereditary dynamical systems is presented. One feature of these systems is that the rate of change of their state depends not only on the current position, like in the classical case, but also on the full path travelled, that is, the history of the motion. Most of the paper is devoted to dynamical systems whose motion is described by functional differential equations of retarded type. In addition, more general systems described by functional differential equations of neutral type and closely related systems described by differential equations with fractional derivatives are considered. So-called path-dependent Hamilton-Jacobi equations are treated, which play for the above classes of systems a role similar to that of the classical Hamilton-Jacobi equations in dynamic optimization problems for ordinary differential systems. In the context of applications to control problems, the main attention is paid to the minimax approach to the concept of a generalized solution of the Hamilton-Jacobi equations under consideration and also to its relationship with the viscosity approach. Methods for designing optimal feedback control strategies with memory of motion history which are based on the constructions discussed are presented. Bibliography: 183 titles.

Minimax solutions of Hamiltoni-Jacobi equations in dynamical optimization problems for hereditary systems

Настоящая статья содержит обзор результатов, касающихся развития теории уравнений Гамильтона-Якоби для наследственных динамических систем. Особенность этих систем состоит в том, что скорость изменения их состояния зависит не только от текущего положения, как в классическом случае, но и от всего пройденного пути - истории движения. Большая часть статьи посвящена динамическим системам, движение которых описывается при помощи функционально-дифференциальных уравнений запаздывающего типа. Кроме того, затрагиваются и более общие системы, описываемые функционально-дифференциальными уравнениями нейтрального типа, а также тесно связанные с ними системы, описываемые дифференциальными уравнениями с производными дробного порядка. Рассматриваются так называемые наследственные уравнения Гамильтона-Якоби, которые для указанных классов систем играют роль, аналогичную роли классических уравнений Гамильтона-Якоби в задачах динамической оптимизации обыкновенных дифференциальных систем. В контексте приложений к задачам управления основное внимание уделяется минимаксному подходу к понятию обобщенного решения рассматриваемых уравнений Гамильтона-Якоби, а также его связи с вязкостным подходом. Приводятся опирающиеся на обсуждаемые конструкции методы построения оптимальных стратегий управления по принципу обратной связи с памятью истории движения. Библиография: 183 названия.

Daily commuting

Workers generally commute on a daily basis, so we model commuting as a repeated game. The folk theorem implies that for sufficiently large discount factors, the repeated commuting game has as a Nash equilibrium any feasible strategy that is uniformly better than the minimax strategy payoff for a commuter in the one shot game, repeated over the infinite horizon. This includes the efficient equilibria. An example where the efficient payoffs strictly dominate the one shot Nash equilibrium payoffs is provided. Our conclusions pose a challenge to congestion pricing in that equilibrium selection could be at least as effective in improving welfare. We examine evidence from St. Louis to determine what equilibrium strategies are actually played in the repeated commuting game.

Negotiation strategies for agents with ordinal preferences: Theoretical analysis and human study

Negotiation is a very common interaction between agents. Many common negotiation protocols work with cardinal utilities, even though ordinal preferences, which only rank the outcomes, are easier to elicit from humans. In this work, we focus on negotiation with ordinal preferences over a finite set of outcomes. We study an intuitive protocol for bilateral negotiations, where the two parties make offers alternately. We analyze the negotiation protocol under two settings: First, we consider the full information setting, where each party is fully aware of the other party's preference order. For this case, we provide elegant strategies that specify a sub-game perfect equilibrium. In addition, we show how the studied negotiation protocol almost completely implements a known bargaining rule. Second, we analyze the complementary no-information setting where neither party knows the preference order of the other party. For this case, we provide a Maxmin strategy and show that every pair of Maxmin strategies specifies a robust-optimization equilibrium. Finally, through a human study (N=150), we empirically study the practical relevance of our full information analysis to people engaging in negotiations with each other and/or with an automated agent using the studied protocol. Surprisingly, our results indicate that people tend to arrive at the equilibrium outcomes despite frequently departing from the proposed strategies. In addition, in contrast to commonly held beliefs, we find that an equilibrium-following agent performs very well with people.

A New Minimax Theorem for Randomized Algorithms

The celebrated minimax principle of Yao says that for any Boolean-valued function f with finite domain, there is a distribution μ over the domain of f such that computing f to error ε against inputs from μ is just as hard as computing f to error ε on worst-case inputs. Notably, however, the distribution μ depends on the target error level ε: the hard distribution which is tight for bounded error might be trivial to solve to small bias, and the hard distribution which is tight for a small bias level might be far from tight for bounded error levels. In this work, we introduce a new type of minimax theorem which can provide a hard distribution μ that works for all bias levels at once. We show that this works for randomized query complexity, randomized communication complexity, some randomized circuit models, quantum query and communication complexities, approximate polynomial degree, and approximate logrank. We also prove an improved version of Impagliazzo’s hardcore lemma. Our proofs rely on two innovations over the classical approach of using Von Neumann’s minimax theorem or linear programming duality. First, we use Sion’s minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as “forecasting algorithms” evaluated by a certain proper scoring rule. The expected score of the forecasting version of a randomized algorithm appears to be a more fine-grained way of analyzing the bias of the algorithm. We show that such expected scores have many elegant mathematical properties—for example, they can be amplified linearly instead of quadratically. We anticipate forecasting algorithms will find use in future work in which a fine-grained analysis of small-bias algorithms is required.

Multidisciplinary design optimization of a dual-spin guided vehicle

In this research, a Multidisciplinary Design Optimization approach is proposed for the dual-spin guided flying projectile design considering external and internal parts of the body as design variables. In this way, a parametric formulation is developed. All related disciplines, including structure, aerodynamics, guidance, and control are considered. Minimum total mass, maximum aerodynamic control effectiveness, minimum miss distance, maximum yield stress in all subsystems, controllability and gyroscopic stability constraints are some of objectives/constraints taken into account. The problem is formulated in All-At-Ones Multidisciplinary Design Optimization approach structure and solved by Simulated Annealing and minimax algorithms. The optimal configurations are evaluated in various aspects. The resulted optimal configurations have met all design objectives and constraints.

Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential

We consider the Lorentz force equationddt(mx˙1−|x˙|2/c2)=q(E(t,x)+x˙×B(t,x)),x∈R3, in the physically relevant case of a singular electric field E. Assuming that E and B are T-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T-periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem.

Data-Driven Cooperative Multi-Task Assignment in Heterogeneous Multi-Agent Pursuit-Evade Game

In this paper, a task assignment method is proposed to deal with the multi-agent pursuit-evade game for heterogeneous unnamed aerial vehicles via reinforcement learning. The mathematical model based on the local position error dynamics is established to describe the interactions among the vehicles in the pursuit-evade game, subject to high nonlinearities and parameter uncertainties involved in the vehicle model. The execution costs and the corresponding optimal control policies of the agent pursuing each target are calculated, and the policy with minimum execution cost is determined as the objective of the multi-agent pursuit-evade game. Min-Max strategy is introduced to estimate and counteract the interaction effects in the mathematical model, and the reinforcement learning-based algorithm is proposed to obtain the optimal solution to the assignment problem based on the Hamilton–Jacobi–Bellman equation without the interaction effects. Simulation results are given to show the effectiveness of the proposed task assignment method.

Development of solar radiation prediction model using minimax algorithm

AbstractRecently, many studies are carried out on solar energy (SE) and global solar radiation (GSR) estimation. When the literature is examined, there are many GSR estimation models developed with different algorithms in different regions of the world. Although some of these models are seen to perform acceptable all over the world, the performance of some models differs according to their location. The Minimax algorithm is a decision‐making algorithm that maximizes the winning potential while minimizing the probability of losing. In this study, a new Angstorm model was developed for GSR with minimax algorithm. The most important aspect of this article in terms of originality is that the GSR estimation model is being developed with the minimax algorithm for the first time. The MATLAB program was used while developing the prediction model with the minimax algorithm. Different statistical error tests were used to evaluate the performance of the results. When these test results are evaluated in general, it is seen that the minimax algorithm gives acceptable results in GSR estimation.

Distributed Minmax Strategy for Consensus Tracking in Differential Graphical Games: A Model-Free Approach

Distributed Minmax Strategy for Consensus Tracking in Differential Graphical Games: A Model-Free Approach

Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac–Coulomb operators

We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property associated with the block decomposition. A typical example is the Dirac–Coulomb operator defined on C^\infty_c(\mathbb{R}^3\setminus\{0\}, \mathbb{C}^4) . In this paper we define a distinguished self-adjoint extension with a spectral gap and characterize its eigenvalues in that gap by a min-max principle. This has been done in the past under technical conditions. Here we use a different, geometric strategy, to achieve that goal by making only minimal assumptions. Our result applied to the Dirac–Coulomb-like Hamitonians covers sign-changing potentials as well as molecules with an arbitrary number of nuclei having atomic numbers less than or equal to 137

Minimax Principle for Eigenvalues of Dual Quaternion Hermitian Matrices and Generalized Inverses of Dual Quaternion Matrices

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency concepts for a set of dual quaternion vectors, and study some related basic properties for dual quaternion vectors and dual quaternion matrices. We present a minimax principle for eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an inequality for singular values of dual quaternion matrices. Finally, we introduce the concept of generalized inverses of dual quaternion matrices, and present necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix.

Normalized solutions for Schrödinger-Poisson equation with prescribed mass: The Sobolev subcritical case and the Sobolev critical case with mixed dispersion

In this paper, we study the following Schrödinger Poisson equation{−Δu−λu−γ(|x|−1⁎|u|2)u=f(u),x∈R3;∫R3u2dx=c, where c>0, λ∈R and f∈C(R,R). When γ<0 and f satisfies some weaker L2-supercritical conditions in Sobolev subcritical case, we show the existence of normalized solutions. When f(u)=μ|u|q−2u+|u|4u with μ>0, γ>0 and 2<q<103, a more complicated situation, we obtain the existence of multiple solutions. Our main tools are some technically simpler than the Ghoussoub minimax principle [18], which not only allow us to construct a bounded (PS) sequence under some weaker conditions on f than before, but help to yield richer results. In particular, our results generalize and improve some ones in [21] and some other related literatures.

Rawls’s maximin and optimal taxation theory

The paper analyses the import and appropriation of Rawls’s theory of justice into the emerging field of optimal taxation theory in the 1970s. It focuses first on the pioneer contributions of Atkinson and Phelps to integrate Rawls’s maximin into optimal taxation models, and then on their numerous followers during this decade. It shows that the maximin criterion was quickly accepted and appropriated in optimal taxation theory using “Rawlsian” Social Welfare Functions, which are founded on a welfarist interpretation of the maximin, unfaithful to Rawls. I try to explain why public economists made this choice, insisting on issues of simplicity, tractability, and comparisons with other ethical criteria.

Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations

In the paper, we consider a path-dependent Hamilton–Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton–Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of inequalities for coinvariant sub- and supergradients. We prove that these two notions are equivalent, which is the main result of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton–Jacobi equation and a right-end boundary condition. The proof of the main result is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein–Preiss variational principle with an appropriate gauge-type functional.

JUSTIFICATION OF APPROACHES BEFORE PLACEMENT OF PASSENGER VESSELS IN THE DIRECTIONS OF THEIR WORK AND IMPROVING THE WORK PLANNING OF THE SHIPPING COMPANY’S FLEET

Abstract. The purpose of the article is to solve the task of arranging the passenger fleet and improving the planning of the fleet of the shipping company. To achieve the goal, the possible operating conditions of the passenger fleet were studied; developed approaches that will allow, taking into account different types of sailing, different directions of work, sizes of ships, as well as the possibility of replenishment of the fleet, to obtain the maximum value of the financial result when deploying the passenger fleet of the shipping company, which made it possible to improve the planning of the passenger fleet of the shipping company. The process of managing a shipping enterprise is a continuous development of management decisions, which largely depends on the final result of the work of any company. This is especially important for shipping, which functions in conditions of significant dynamics of both the external environment and the direct production of transport processes. The proposed solution will allow the shipping company to effectively locate its fleet, obtain the maximum financial result, taking into account the conditions of transportation and operation of vessels of various types of navigation, forms of organization of transportation, various types of vessels, routes, and also takes into account the possibility of replenishing the fleet of the shipping company in the event that its potential freight capacity does not exceed the amount of transport work. One of the most important areas of the shipping company’s work is planning and optimizing the use of the company’s resources. The shipping company strives for the maximum financial result from the work performed. If we consider the "external environment" as a management subject that also makes decisions corresponding to the options of the possible structure of passenger flows, then this kind of task can be formalized and presented in the form of a matrix game. A characteristic condition is that each party has no information about the decision of the other. Thus, the maximin (minimax) principle is implemented in a conflict situation. The value of the game determines the mathematical expectation of winning the first player, that is, the value of the financial result, which should be taken as planned. As a result, the forecast of the results of the shipping company will reflect the production capabilities of the "external environment" (the second player) and the conditions in which it will have to work. The proposed approach can also be used to determine the values of the planned indicators of other maritime transport enterprises - ports, ship repair plants, as well as standards of transport work; development of the replenishment (write-off) program of the shipping company’s fleet; determining the tariff policy related to the transportation of passengers in the shipping company’s region.

Robust Fusion Kalman Estimator of the Multi-Sensor Descriptor System with Multiple Types of Noises and Packet Loss.

Under the influence of multiple types of noises, missing measurement, one-step measurement delay and packet loss, the robust Kalman estimation problem is studied for the multi-sensor descriptor system (MSDS) in this paper. Moreover, the established MSDS model describes uncertain-variance noises, multiplicative noises, time delay and packet loss phenomena. Different types of noises and packet loss make it more difficult to build the estimators of MSDS. Firstly, MSDS is transformed to the new system model by applying the singular value decomposition (SVD) method, augmented state and fictitious noise approach. Furthermore, the robust Kalman estimator is constructed for the newly deduced augmented system based on the min-max robust estimation principle and Kalman filter theory. In addition, the given estimator consists of four parts, which are the usual Kalman filter, predictor, smoother and white noise deconvolution estimator. Then, the robust fusion Kalman estimator is obtained for MSDS according to the relation of augmented state and the original system state. Simultaneously, the robustness is demonstrated for the actual Kalman estimator of MSDS by using the mathematical induction method and Lyapunov's equation. Furthermore, the error variance of the obtained Kalman estimator is guaranteed to the upper bound for all admissible uncertain noise variance. Finally, the simulation example of a circuit system is examined to illustrate the performance and effectiveness of the robust estimators.

Physics-informed learning under epistemic uncertainty with an application to system health modeling

This study proposes a methodology for developing deterioration models to estimate the remaining lifetime of a system using physics-informed learning. The approach consists of combining physical knowledge about the system with a set of data obtained from similar systems that have failed in the past to build a set of constraints on the evolution over time of the state of health of the system. The data consists of partial measurements of the system variables, taken from its commissioning to the present. The physical knowledge used comprises a set of differential equations that approximate the dynamics and aging of the system. In contrast to other studies, the physical model is not assumed to be accurate, but is considered to be an approximation of reality. Constraints on the temporal evolution of the state of health derived from this physical knowledge take into account its imprecision and consist of possibility distributions. In this study, the max-max, max-min and min-max regret principles are applied to extract the time evolution of the deterioration rate that best fits the constraints. The effectiveness of the proposed algorithm is evaluated by means of a comparative empirical analysis in different use cases, and the situations in which informed learning improves on purely data-driven algorithms are analyzed.

Zero-Sum Games and Linear Programming Duality

The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig is known to be incomplete. We explain and combine classical theorems about solving linear equations with nonnegative variables to give a correct alternative proof more directly than Adler. We also extend Dantzig’s game so that any max-min strategy gives either an optimal LP solution or shows that none exists.