Minimax Procedure Research Articles

Equivariant index bound for min–max free boundary minimal surfaces

Given a three-dimensional Riemannian manifold with boundary and a finite group of orientation-preserving isometries of this manifold, we prove that the equivariant index of a free boundary minimal surface obtained via an equivariant min–max procedure á la Simon–Smith with n-parameters is bounded above by n.

Normalized saddle solutions for a mass supercritical Choquard equation

We investigate the existence of saddle type normalized solutions for the nonlinear Choquard equation:{−Δu−λu=(Iα⁎F(u))F′(u), in RN∫RN|u|2dx=a2,u∈H1(RN). Here N≥1, a>0 is given in advance, Iα is the Riesz potential of order α∈(0,N) and the unknown parameter λ appears as a Lagrange multiplier. In a mass supercritical setting on F, we prove the existence of a couple (uaG,λaG)∈H1(RN)×R− of saddle solutions for any a>0 and for given finite Coxeter group G with its rank k≤N. Our method is to combine the concentration compactness principle with a minimax procedure in the saddle type symmetric subspace, which gives a variational framework of constructing normalized saddle solutions for the Choquard equation.

The phenomenon of decision oscillation: A new consequence of pathology in game trees

AbstractRandom minimaxing studies the consequences of using a random number for scoring the leaf nodes of a full width game tree and then computing the best move using the standard minimax procedure. Experiments in Chess showed that the strength of play increases as the depth of the lookahead is increased. Previous research by the authors provided a partial explanation of why random minimaxing can strengthen play by showing that, when one move dominates another move, then the dominating move is more likely to be chosen by minimax. This paper examines a special case of determining the move probability when domination does not occur. Specifically, we show that, under a uniform branching game tree model, whether the probability that one move is chosen rather than another depends not only on the branching factors of the moves involved, but also on whether the number of ply searched is odd or even. This is a new type of game tree pathology, where the minimax procedure will change its mind as to which move is best, independently of the true value of the game, and oscillate between moves as the depth of lookahead alternates between odd and even.

Linear Combination of Leukocyte Count and D-Dimer Levels in the Diagnosis of Patients with Non-traumatic Acute Abdomen

Aim: Rapid intervention is required in patients with non-traumatic acute abdominal pain. It is very important to distinguish between surgical and non-surgical pathologies during this intervention. This study aimed to increase the diagnostic accuracy by combining the leukocyte count and D-dimer levels used in this evaluation with linear combination methods. Materials and Methods: Logistic regression, scoring, min-max, minimax, Su & Liu, Pepe & Thompson, Pepe, Cai & Langton, and Todor & Saplacan methods were used as linear combination methods. The data set was divided into 70% training set and 30% test set. Parameter optimization was performed on the training data by 5 fold cross-validation method using 10 repeats. The area under the ROC curve, sensitivity, selectivity, accuracy, positive and negative predictive value, and positive and negative likelihood ratio statistics were used in the performance evaluation.Results: The area under the ROC curve statistic for D-dimer level and log-transformed leukocyte count variable were obtained as 0.71 and 0.70, respectively. The accuracy rate was 0.69 for the D-dimer level and 0.73 for log-transformed leukocyte count. For the linear combination methods, the area under the ROC curve was between 0.77 and 0.81, and the accuracy statistics were between 0.72 and 0.79. The best performance was obtained with the min-max method.Conclusion: In patients with non-traumatic acute abdominal pain, leukocyte count and D-dimer levels can be evaluated together by using linear combination methods in differentiating surgical and non-surgical pathologies. The obtained results showed that the diagnostic performance of the combined results with the min-max procedure was higher than the leukocyte count and D-dimer levels.

Weighted shrinkage estimators of normal mean matrices and dominance properties

In the estimation of the mean matrix in a multivariate normal distribution, the Efron–Morris estimator and the James–Stein estimator are two well-known minimax procedures, where the former is matricial shrinkage and the latter is scalar shrinkage. The methods for combining the two estimators with random weight functions are addressed. For deriving weight functions, the paper suggests the two methods. One is the minimization of a part of the unbiased estimator of the risk function, and the other is the empirical Bayes approach. The resulting weighted shrinkage estimators are shown to be minimax, and the extension to the case of an unknown covariance matrix is developed.

Nonparametric regression with responses missing at random and the scale depending on auxiliary covariates

Nonparametric regression with missing at random (MAR) responses, univariate regression component of interest, and the scale function depending on both the predictor and auxiliary covariates, is considered. The asymptotic theory suggests that both heteroscedasticity and MAR mechanism affect the sharp constant of the minimax mean integrated squared error (MISE) convergence. Our sharp minimax procedure is based on the estimation of unknown nuisance scale function, design density and availability likelihood. The estimator is adaptive to the missing mechanism and unknown smoothness of the estimated regression function. Simulation studies and real examples also justify practical feasibility of the proposed method for this complex regression setting.

A predictive fault tolerant control method for qLPV systems subject to input faults and constraints

In this paper, we investigate the use of Model Predictive Control (MPC) applications for quasi-Linear Parameter Varying (qLPV) systems subject to faults along the input channels. We propose a Fault Tolerant Control (FTC) mechanism based on a robust state-feedback MPC synthesis, considering polytopic inclusions. In order to alleviate the numerical burden of the robust min-max procedure, we use small prediction horizons, in such a way that the solution becomes viable for real-time systems. The FTC system is able to tolerate time-varying saturation of the actuator, which may happen due to malfunctions. Recursive feasibility and poly-quadratic stability guarantees are ensured through the synthesis of adequate terminal ingredients. Accordingly, we present a catalogue of three different LMI remedies, considering: (a) parameter-independent ingredients, (b) a parameter-dependent terms and (c) a parameter-dependent maps that take into account bounded rates of parameter variation. An autonomous driving car example is used to illustrate the performances of the proposed technique, which is compared to other MPCs from the literature. The proposed FTC method is able to ensure good performances, obtained with reduced computational demand.

Digital Trade Negotiation in Korea-China-Japan FTA: Minimax Strategy in the Process of Making Consensus

Purpose – The purpose of this paper is to apply the minimax procedures of the negotiation model to digital trade negotiation in the KCJ FTA to predict possible negotiation outcomes, as well as sensitive issues, for each country. Design/Methodology/Approach – Based on the negotiation model of the minimax procedures, this paper first derives top preferences for each country in negotiating the KCJ FTA. It then conducts a sectional analysis of issues and derives possible negotiation outcomes in terms of both majority voting (MV) outcomes and minimax outcomes (FBn). It finally predicts sensitive issues in the voting process and outcome rankings for each section of issues. Findings – For the first section of issues, the minimax outcome will prevail, while the MV outcome will be the first in ranking for the last section of issues. In terms of sensitivity during the negotiation process for voting, China is predicted to be the most reluctant to agree on binding rules on non-discrimination. As Japan is not as sensitive as the other two countries in each issue of digital trade agreement, both Korea and China are predicted to be the most hesitative in agreeing on binding rules on source code. Research Implications – Under the minimax procedures, negotiation positions as well as the dynamics of the voting process for each issue can be systematically predicted ex ante. The negotiation outcomes derived from the model can be viewed as win-win that would not produce any left-out countries, since each country minimizes the deviation from the initial top preference to the agreed deal.

Accurate calculations for the Dirac electron in the field of two-center Coulomb field: Application to heavy ions

The relativistic Dirac equation for a bound electron in the field of two fixed positive charges is revisited. In contrast to the one center case this three dimensional equation is separable only partially around the azimuthal angle φ, because of the commutation of the Dirac Hamiltonian only with the z component of the total angular momentum Jz. In this work we determine the variational exact solution of this two center problem using a basis constructed by linear combinations of relativistic Slater type spinor wave functions with non integer powers of the radii r1 and r2 on the two centers. We present in some detail the determination of the two center integrations involved. The solutions are obtained by a minimax procedure, that we have developed with a new iterative scheme. We use independent large and small components of the Dirac spinor. This permits us to take control of the spurious solutions, and gives us the possibility to avoid them by the appropriate choice of the wave function parameters. We investigate the behavior of the electron in its 1sσg level of the diatomic homo-nuclear systems A2(2Z-1)+, where A represents the heavy element and Z its atomic number. In the case of heavy ions we study the dependance of the electron energy on the internuclear distance, this gives us an indication for the conditions of atomic collapse, which can induce electron-positron pair production. Our approach has the advantage of needing small basis sets for a relative error of the of 10-7-10-8. It can also be extended easily to the excited level such as the 1sσu level.

Allen–Cahn min-max on surfaces

We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage to study the fine structure of potential singular points at the diffuse level, and show that the problem reduces to that of understanding "entire" singularity models constructed by del Pino-Kowalczyk-Pacard with Morse index 1. The argument is completed by a Morse index estimate on these singularity models.

Stochastic multi-step saddle point process minimax control parametric synthesis

The actual research problem of operations is development of methods of increase of a management efficiency by processes of the conflict nature. Article is devoted to development of methods of increase of a management efficiency by such processes. The purpose of article is the substantiation of the approach to parametrical synthesis of optimum control by multi-step stochastic minimax processes and procedures of the numerical analysis of likelihood dynamic characteristics of process. Formalization of process consists in definition of its type, a vector of phase coordinates and corresponding restrictions, the task of set of the actions sold by each the parties, efficiency of each action, control parameters (varied parameters of process) which task of values each of the parties influences a course of process, control restrictions, criteria of efficiency of the parties expressed through elements of a vector of phase coordinates. Discrete final stochastic process is considered. Change of phase coordinates occurs during the discrete moments of time, named steps of process. Phase coordinates depend on values of two groups of control parameters (controls of the counteracting parties). Within the limits of the modern theory of optimization of stochastic systems procedure of synthesis of optimum control is realized two-phase. At the first stage with use of analytical methods the structure of optimum control is determined. For these purposes the simplified determined model of process can to be used. At the second stage parametrical control optimization with use of algorithmic methods and computing procedures statistical linearization is carried out. Dynamics of process is described vectorial finite-difference equation. It is necessary to distinguish cases when there is saddle a point and when saddle the point is absent. Parametrical synthesis of optimum control is possible only in the first case. It is considered three basic variants of the equation: the linear equation; the nonlinear equation with optimum controls on border of a range of definition; the nonlinear equation with optimum controls inside of a range of definition. For the first variant there is an effective algorithm of parametrical synthesis of optimum control. For the second variant of synthesis of optimum control it is possible, but the algorithm is not effective. For the third variant to determine optimum managements it is not possible. Procedure statistical linearization is offered. Procedure consists in generation of set of realizations of the casual process set by the vector equation, calculation of optimum control for each concrete realization and the further statistical processing of the received results. The process described by the piecewise linear vector equation, is a special case of nonlinear process. At that it keeps property of independence of optimum control from coordinates of process. It provides expansion of a scope of effective computing procedure of synthesis of optimum control on a new class of piecewise linear processes. Property of a constancy process Hamiltonian can be used as criterion of correctness of calculation of optimum control in concrete cases. Application of the offered procedure provides use of methods of statistical modelling for the decision of tasks of the analysis of dynamics of the conflict and synthesis of optimum control in view of nonlinearity of functions of losses of the parties, dependence of efficiency of means used by them on the random factors formalized in the form of stochastic functions with various likelihood distributions, and also uncertainty concerning actions of the opponent.

Willmore minmax surfaces and the cost of the sphere eversion

We develop a general minmax procedure in Euclidian spaces for constructing Willmore surfaces of non-zero indices. We apply this procedure to the Willmore minmax sphere eversion in the 3-dimensional Euclidian space. We compute the cost of sphere eversion in terms of Willmore energies of the Willmore spheres in \mathbb R^3 .

The catenoid estimate and its geometric applications

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the “doublings” of the Clifford torus by Kapouleas–Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved by an index $1$ orientable minimal hypersurface.

A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space

We establish a version of the Trudinger-Moser inequality involving unbounded or decaying radial weights in weighted Sobolev spaces. In the light of this inequality and using a minimax procedure we also study existence of solutions for a class of quasilinear elliptic problems involving exponential critical growth.

Infinitely many solutions for a class of critical Choquard equation with zero mass

In this paper we investigate the following nonlinear Choquard equation $$ -\Delta u =\bigg(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}dy\bigg)g(x,u)\quad \textrm{in} \mathbb{R}^N, $$ where $0< \mu< N$, $N\geq3$, $g(x,u)$ is of critical growth in the sense of the Hardy-Littlewood-Sobolev inequality and $G(x,u)=\int^u_0g(x,s)ds$. By applying minimax procedure and perturbation technique, we obtain the existence of infinitely many solutions.

On Equations of the Program Iteration Method

The program iteration method, which is inseparably associated with the name of A. G. Chentsov, first appeared in the study of the so-called zero-sum differential games. At early stages, only one of the two possible dual iterative procedures was considered — the maximin procedure. This can be explained by the special interest of the researchers in the so-called program maximin function, which is conveniently interpreted in geometric terms in games of pursuit. Nevertheless, the dual minimax iterative procedure is of no less interest. The program iteration method is mainly significant because it may be used as the basis for the development of a differential game theory in a closed compact form, which was shown earlier for a version of the method based on a certain modification of iterative operators. The key role in this theory belongs to the theorem that states the existence and uniqueness of a solution of the equation induced by a pair of such operators. In this case, the maximin iterative procedure is used to describe e-optimal (in some cases, optimal) positional strategies of the first player, while the minimax procedure is used to describe e-optimal (in some cases, optimal) positional strategies of the second player. This paper investigates the structure of the solution set of the generalized Bellman-Isaacs equation obtained with the use of historically first (not modified) operators of the program iteration method. A theorem that states the existence and uniqueness of the solution to this equation meeting a natural boundary condition is proved under certain assumptions. Thus, it is shown that the original version of the program iteration method can also be used in designing a closed-form differential game theory. However, here we use the so-called recursive strategies rather than positional ones. Such strategies, together with the program iteration method, play an essential role in the analysis of coalition-free differential games.

An Iterated Min–Max procedure for practical workload balancing on non-identical parallel machines in manufacturing systems

This paper presents an original approach for a practical workload balancing problem on non-identical parallel machines in manufacturing systems. After showing the limitations of an initial model, in particular to support relevant decisions, the min–max fairness workload balancing problem is motivated and positioned in the literature. The Iterated Min–Max (IMM) procedure is then presented, with its properties, and illustrated. The IMM consists in solving a succession of linear programs using information from dual variables obtained at each iteration. Computational results on industrial instances show the relevance of the approach when compared to the initial model. The current use of the IMM procedure in an industrial tool is discussed.

Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

In this paper, we study the existence and multiplicity of solutions with a prescribed ${L^2}$-norm for a class of nonlinear fractional Choquard equations in ${\R^N}$: where $N\geq3$, $s\in~(0,1)$, $\alpha~\in(0,N)$, $p\in(\max~\{~1~+~\frac{{\alpha~+~2s}}{N},2\},\frac{N+\alpha}{N-2s})$ and ${\kappa~_\alpha~}(x)~=~{~|~x~|^{\alpha~-~N}}$.To get such solutions, we look for critical points of the energy functional on the constraints 0. \end{equation}]]> For the value $p\in(\max~\{~1~+~\frac{{\alpha~+~2s}}{N},2\},\frac{N+\alpha}{N-2s})$ considered, the functional $I$ is unbounded from below on $S(c)$. By using the constrained minimization method on a suitable submanifold of $S(c)$, we prove that for any $c>0$, $I$ has a critical point on $S(c)$ with the least energy among all critical points of $I$ restricted on $S(c)$. After that, we describe a limiting behavior of the constrained critical point as $c$ vanishes and tends to infinity. Moreover, by using a minimax procedure, we prove that for any $c>0$, there are infinitely many radial critical points of $I$ restricted on $S(c)$.

Controlling the vibration process of vibration protection systems with dynamic damping

The article deals with theoretical aspects of controlling the oscillation process in vibration protection systems with a dynamic damping based on the use of modern information technologies. The justification of the principle of minimum and minimax procedure for the formation of optimal control of oscillation processes is given. It is shown that a direct method of integrating the equations of state, when observing the minimax procedure, allows us to find the values of the components of the optimal control vector directly at each integration step. The minimax procedure algorithm of the minimum principle is used to solve the optimization problem of dynamic damping of oscillations. A synthesizing control function is found that allows eliminating resonant phenomena and providing attenuation of transient processes within a single period of kinematic action.

Entropy of closed surfaces and min-max theory

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding–Ilmanen–Minicozzi–White conjectured (since proved by Bernstein–Wang) that the entropy of any closed surface is at least that of the self-shrinking two-sphere. In this paper we give an alternative proof of their conjecture for closed embedded 2-spheres. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques–Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in $\mathbb{R}^3$ which we also establish. To any closed surface in $\mathbb{R}^3$ we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for $\mathbb{R}^3$ with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of nonstationary varifolds (with respect to the Gaussian metric of $\mathbb{R}^3$) in a continuous manner.