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We analyze the semi‐implicit scheme of high‐index saddle dynamics, which provides a powerful numerical method for finding the any‐index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi‐implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi‐implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi‐implicit scheme via, for example, technical splittings and multi‐variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi‐implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi‐implicit scheme in constructing solution landscapes of complex systems. [ABSTRACT FROM AUTHOR]

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating the primal problem NIMSIPVC and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints. [ABSTRACT FROM AUTHOR]

There is considered the possibility of solving the problem of choosing the optimal width of curved emergency corridors, both sides of which are parts of an ellipse to further minimize the mixing of human flows (and, accordingly, to prevent congestion in people's movement) during evacuation in case of fire or other emergencies. The issue of changing the corridor width along its length is considered in two ways: one-dimensional unconditional optimization and the classical Kuhn-Tucker theorem. A sequence of steps is proposed to implement the task of finding the optimal corridor width.

In this paper, we introduce the concept of generalized saddle points for a function defined on a product of topological spaces. The generalized saddle points are defined as minima of a special function, that is the supremum of Nikaido-Isoda function. We prove elementary properties of the generalized saddle points and study their dependence on a parameter. We show that under natural assumptions if a function has the only generalized saddle point, then this point is stable. Moreover, for functions defined over a sets of solutions to nonlinear equations with a parameter in finite-dimensional spaces, we provide sufficient conditions of the generalized saddle point stability in terms of λ-truncations. [ABSTRACT FROM AUTHOR]

A quadratic minimization problem is considered in Hilbert spaces under constraints given by a linear operator equation and a convex quadratic inequality. The main feature of the problem statement is that the practically available approximations to the exact linear operators specifying the criterion and the constraints converge to them only strongly pointwise rather than in the uniform operator norm, which makes it impossible to justify the use of the classical regularization methods. We propose a regularization method that is applicable in the presence of error estimates for approximate operators in pairs of other operator norms, which are weaker than the original ones. For each of the operators, the pair of corresponding weakened operator norms is obtained by strengthening the norm in the domain of the operator and weakening the norm in its range. The weakening of operator norms usually makes it possible to estimate errors in operators where this was fundamentally impossible in the original norms, for example, in the finite-dimensional approximation of a noncompact operator. From the original optimization formulation, a transition is made to the problem of finding a saddle point of the Lagrange function. The proposed numerical method for finding a saddle point is an iterative regularized extragradient two-stage procedure. At the first stage of each iteration, an approximation to the optimal value of the criterion is refined; at the second stage, the approximate solution with respect to the main variable is refined. Compared to the methods previously developed by the authors and working under similar information conditions, this method is preferable for practical implementation, since it does not require the gradient step size to converge to zero. The main result of the work is the proof of the strong convergence of the approximations generated by the method to one of the exact solutions to the original problem in the norm of the original space. [ABSTRACT FROM AUTHOR]

The effect of viscosity on nonrelativistic oscillations of cold plasma is numerically analyzed. For this purpose, an implicit difference scheme of the MacCormack type is constructed, which has a weaker restriction on stability in comparison with the explicit scheme. The scheme is implemented without iterations, which increases its computational efficiency tenfold. It is shown that taking into account the plasma viscosity can cause not only attenuation of the amplitude of plasma oscillations, but also a change in the shape of the solution. With an increase in the viscosity coefficient, a saddle point is observed in the solution, which is preserved in time. [ABSTRACT FROM AUTHOR]

The minimum (worst case) value of a long-only portfolio of bonds, over a convex set of yield curves and spreads, can be estimated by its sensitivities to the points on the yield curve. We show that sensitivity based estimates are conservative, i.e., underestimate the worst case value, and that the exact worst case value can be found by solving a tractable convex optimization problem. We then show how to construct a long-only bond portfolio that includes the worst case value in its objective or as a constraint, using convex–concave saddle point optimization. [ABSTRACT FROM AUTHOR]

We propose a novel passive micromixer leveraging STC (split-to-circulate) flow characteristics and analyze its mixing performance comprehensively. Three distinct designs incorporating submerged circular walls were explored to achieve STC flow characteristics, facilitating flow along a convex surface and flow impingement on a concave surface. Across a broad Reynolds number range (0.1 to 80), the present micromixer substantially enhances mixing, with a degree of mixing (DOM) consistently exceeding 0.84. Particularly, the mixing enhancement is prominent within the low and intermediate range of Reynolds numbers ( 0.1 < R e < 20) . This enhancement stems from key flow characteristics of STC: the formation of saddle points around convex walls and flow impingement on concave walls. Compared to other passive micromixers, the DOM of the present micromixer stands out as notably high over a broad range of Reynolds numbers ( 0.1 ≤ R e ≤ 80 ). [ABSTRACT FROM AUTHOR]

In this paper, the ignition boundary and height of the saddle point as the lowest heating power on the way to ignition for D-3He toroidal magnetic configuration are derived. Fusion fuel cycle involving deuterium and helium-3, along with side reactions is considered. The effect of fuel dilution, incorporating synchrotron radiation and bremsstrahlung power loss with relativistic correction on ignition criteria, is investigated. The results were compared by considering the classical state of bremsstrahlung power. It is found that the height of the saddle point is increased with increasing the ratio of the proton particle density to the fuel deuteron density. The ignition regime shrinks and the minimum value of the Lawson parameter for the breakeven condition and ignition boundaries is reduced due to fuel dilution.

This work deals with a saddle point formulation of parameter identification in linear elastic contact problems with friction. Using the primal–dual formulation of the constrained minimization problem and given observations, we estimate the Lamé coefficients through the penalization and dualization of the considered inverse problem. By Fenchel duality, we provide the dual energy function associated with the constraint. We prove the existence of a solution to the regularized parameter identification problem as well as the convergence of the penalized problem to the original one. An augmented Lagrangian formulation of the inverse problem and the existence of its saddle point are provided. By means of the alternating direction method of multipliers (ADMM) and a primal–dual active set strategy (PDAS), we solve the problem numerically and illustrate our approach.

In this paper, we propose a new algorithm for solving convex-concave saddle-point problems using regret minimization in the repeated game framework. To do so, we introduce the Conic Blackwell Algorithm+ (CBA+), a new parameter- and scale-free regret minimizer for general convex compact sets. CBA+ is based on Blackwell approachability and attains O(T) regret. We show how to efficiently instantiate CBA+ for many decision sets of interest, including the simplex, ℓp norm balls, and ellipsoidal confidence regions in the simplex. Based on CBA+ , we introduce SP-CBA+ , a new parameter-free algorithm for solving convex-concave saddle-point problems achieving a O(1/T) ergodic convergence rate. In our simulations, we demonstrate the wide applicability of SP-CBA+ on several standard saddle-point problems from the optimization and operations research literature, including matrix games, extensive-form games, distributionally robust logistic regression, and Markov decision processes. In each setting, SP-CBA+ achieves state-of-the-art numerical performance and outperforms classical methods, without the need for any choice of step sizes or other algorithmic parameters. Funding: J. Grand-Clément is supported by the Agence Nationale de la Recherche [Grant 11-LABX-0047] and by Hi! Paris. C. Kroer is supported by the Office of Naval Research [Grant N00014-22-1-2530] and by the National Science Foundation [Grant IIS-2147361]. [ABSTRACT FROM AUTHOR]

The current study investigates to characterize the Complex Programming Problem (CPP) solution in a fuzzy environment. The paper is divided into two parts: 1) the first presents a Fuzzy Complex Programming Problem (FCPP) with fuzzy complex constraints, and 2) the second presents the optimality criteria using the fuzzy complex cone. The CPP is suggested by involving fuzzy numbers in the constraints in parts. Using the α -cut set concepts, the problem is converted into the α -complex programming. A number of basic theorems with proofs are established concerning the basic results for the fuzzy complex set of solutions for the F-CPP, and the optimality criteria of the saddle point for F-CPP with fuzzy cones is derived. [ABSTRACT FROM AUTHOR]

An attack–defense model is considered in which the attack can deliver several strikes on defense points to maximize the damage. A way of creating a solution to a zero-sum game in mixed strategies is developed. [ABSTRACT FROM AUTHOR]

In this study, sufficient optimality criteria for a robust fractional interval-valued optimization problem (RP) is highlighted based on the idea of convexificators. The use of generalized invexity tools to support sufficient optimality conditions is demonstrated by an example. Also, saddle point condition of a Lagrange function is examined for RP. Furthermore, Mond–Weir-type robust dual model is constructed and by employing the concept of generalized invexity, usual duality results are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Salkuyeh et al. (D.K. Salkuyeh, H. Aslani, Z.Z. Liang, An alternating positive semi-definite splitting preconditioner for the three-by-three block saddle point problems, Math. Commun. 26 (2021) 177-195) has recently established an alternating positive semi-definite splitting (APSS) method for nonsymmetric block three-by-three nonsingular saddle point problems arising from the Picard iteration method for a class of mixed finite element scheme. In this work, we analyse the semi-convergence of the APSS method for solving a class of nonsymmetric block three-by-three singular saddle point problems. The APSS induced preconditioner is applied to improve the semi-convergence rate of the exible GMRES (FGMRES) method. Numerical results are designated to support the theoretical results. These results show that the served preconditioner is efficient compared with FGMRES without a preconditioner. [ABSTRACT FROM AUTHOR]

Constrained extremal points of a potential can be represented as saddle points of a Lagrangian. We prove existence of such saddle points for nonquadratic potentials in Banach spaces under nonlinear constraints. We then apply these results to the so-called mixed formulation of nonlinear elliptic partial differential equations. [ABSTRACT FROM AUTHOR]

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating the primal problem NIMSIPVC and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.

We propose a novel passive micromixer leveraging STC (split-to-circulate) flow characteristics and analyze its mixing performance comprehensively. Three distinct designs incorporating submerged circular walls were explored to achieve STC flow characteristics, facilitating flow along a convex surface and flow impingement on a concave surface. Across a broad Reynolds number range (0.1 to 80), the present micromixer substantially enhances mixing, with a degree of mixing (DOM) consistently exceeding 0.84. Particularly, the mixing enhancement is prominent within the low and intermediate range of Reynolds numbers (0.1

We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraint, which significantly complicate the numerical scheme and the corresponding numerical analysis. Furthermore, as the existing numerical analysis results usually depend on the index of the saddle points implicitly, the proved numerical accuracy may be reduced if the index is high in many applications, which indicates the lack of robustness with respect to the index. To address these issues, we derive the error estimates for numerical discretization of the constrained high-index saddle dynamics on the high-dimensional sphere, and then improve it by providing an index-robust error analysis in an averaged norm by adjusting the relaxation parameters. The developed results provide mathematical supports for the accuracy of numerical computations. [ABSTRACT FROM AUTHOR]

This paper introduces an average-value-at-risk (AVaR) criterion for discrete-time zero-sum stochastic games with varying discount factors. The state space is a Borel space, the action space is denumerable, and the payoff function is allowed to be unbounded. We first transform the AVaR game problem into a bi-level optimization-game problem in which the outer optimization problem is a problem of minimizing a function of a single variable and the inner game problem has been shown to be equivalent to a so-called expected-discounted-positive-deviation (EDPD) game for discrete-time stochastic game. We solve the EDPD game problem in advance. More precisely, under suitable conditions, we not only establish the Shapley equation, the existence of the value of the game, and saddle points, but also prove that the saddle points can be computed by introducing a primal linear program and a dual linear program. Then, we show that the outer problem can be settled by solving the EDPD game problem. Furthermore, we provide an algorithm for computing (or at least approximating) the value of the game and the saddle points for the AVaR game problem. Finally, as an application, we apply our main results to an inventory-production system with numerical experiments. [ABSTRACT FROM AUTHOR]

In this study, the behaviour of the triple product ( n i τ E T i ) and height of the saddle point are investigated on the basis of the particle and power balance equations by considering primary reaction and side reactions among all product and reactant species in a spherical torus nuclear fusion reactor. For this purpose, a set of equations are coupled and solved numerically. Ignition capability is investigated and compared for different types of side reactions and other features like, the ratio between particle and energy confinement time and hot ion modes. Finally, the results are compared with each other to improve future research. [ABSTRACT FROM AUTHOR]

A parametric version of the ellipsoid method with space scaling by a scalar parameter λ > 0 is considered. For certain values of the parameter λ, it is reduced to the well-known versions of the ellipsoid method, namely, the Shor, Khachiyan, Nemirovsky, and Yudin ones. The properties of two algorithmic implementations of the parameterized ellipsoid method are proved. The first algorithm is based on updating the asymmetric matrix B, and the second one updates the symmetric matrix H = BBT. The applications of the algorithms for solving convex programming problems and finding the saddle point of a convex-concave function are described. [ABSTRACT FROM AUTHOR]

The authors prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere. Due to the semi-implicit treatment and the novel computational procedure, the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations, and the computations of the state variable and directional vectors are coupled with the retraction, the vector transport and the orthonormalization procedure, which significantly complicates the analysis. They address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings. [ABSTRACT FROM AUTHOR]

A parametric version of the ellipsoid method with space scaling by a scalar parameter λ>0 is considered. For certain values of the parameter λ, it is reduced to the available versions of the ellipsoid method, namely, the Shor, Khachiyan, Nemirovsky, and Yudin ones. The properties of two algorithmic implementations of the parameterized ellipsoid method are proved. The first algorithm is based on updating the asymmetric matrix B, and the second one updates the symmetric matrix H= BBT. The applications of the algorithms for solving convex programming problems and for finding the saddle point of a convex-concave function are described. [ABSTRACT FROM AUTHOR]

We propose a model-free shrinking-dimer saddle dynamics for finding any-index saddle points and constructing the solution landscapes, in which the force in the standard saddle dynamics is replaced by a surrogate model trained by the Gassian process learning. By this means, the exact form of the model is no longer necessary such that the saddle dynamics could be implemented based only on some observations of the force. This data-driven approach not only avoids the modeling procedure that could be difficult or inaccurate, but also significantly reduces the number of queries of the force that may be expensive or time-consuming. We accordingly develop a sequential learning saddle dynamics algorithm to perform a sequence of local saddle dynamics, in which the queries of the training samples and the update or retraining of the surrogate force are performed online and around the latent trajectory in order to improve the accuracy of the surrogate model and the value of each sampling. Numerical experiments are performed to demonstrate the effectiveness and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we aim to solve a convex-concave min–max optimization problem, where the convex-concave coupling function is nonsmooth in both variables. We propose a simple subgradient method which is simultaneously updating the iterates through their delayed subgradients at stale iterates of their own variable together with the current iterates of the other variable. We investigate the convergence behavior of the proposed method by providing an upper bound of the function value at the coupling averaged iterates to the saddle value. Specifically, the obtained result refers to the function value at the coupling averaged iterates as the function value of an approximate saddle point at the current iteration. The numerical results demonstrate that, with a suitable strategy of delays’ selections, the delayed subgradient method can achieve better convergence behaviors than its non-delayed counterpart.

This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz's function. Furthermore, a kind of approximate pseudo/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems. [ABSTRACT FROM AUTHOR]

In this paper, we study min-max optimization problems on Riemannian manifolds. We introduce a Riemannian Hamiltonian function, minimization of which serves as a proxy for solving the original min-max problems. Under the Riemannian Polyak-Lojasiewicz condition on the Hamiltonian function, its minimizer corresponds to the desired min-max saddle point. We also provide cases where this condition is satisfied. For geodesic-bilinear optimization in particular, solving the proxy problem leads to the correct search direction towards global optimality, which becomes challenging with the min-max formulation. To minimize the Hamiltonian function, we propose Riemannian Hamiltonian methods (RHMs) and present their convergence analyses. We extend RHMs to include consensus regularization and to the stochastic setting. We illustrate the efficacy of the proposed RHMs in applications such as subspace robust Wasserstein distance, robust training of neural networks, and generative adversarial networks. [ABSTRACT FROM AUTHOR]

Minimax problems of the form minx maxy Ѱ(x, y) have attracted increased interest largely due to advances in machine learning, in particular generative adversarial networks and adversarial learning. These are typically trained using variants of stochastic gradient descent for the two players. Although convex-concave problems are well understood with many efficient solution methods to choose from, theoretical guarantees outside of this setting are sometimes lacking even for the simplest algorithms. In particular, this is the case for alternating gradient descent ascent, where the two agents take turns updating their strategies. To partially close this gap in the literature we prove a novel global convergence rate for the stochastic version of this method for finding a critical point of Ѱ(.):= maxy Ѱ (.,y) in a setting which is not convex-concave. [ABSTRACT FROM AUTHOR]

The paper proves the minimax theorem for a specific class of functions that are defined on branching polylines in a linear space, not on convex subsets of a linear space. The existence of a saddle point for such functions does not follow directly from the classical minimax theorem and needs individual consideration based both on convex analysis and graph theory. The paper presents a self-sufficient analysis of the problem. It contains everything that enables a plain understanding of the main result and its proof and avoids using concepts outside the scope of obligatory mathematical education of engineers. The paper is addressed to researchers in applied mechanics, engineering, and other applied sciences as well as to mathematicians who lecture convex analysis and optimization methods to non-mathematicians. [ABSTRACT FROM AUTHOR]

In this work, we use the notion of convexificators to discuss optimality conditions for a fractional interval-valued optimization problem. We illustrate the sufficient optimality conditions established in the paper by the example of a nonconvex fractional interval-valued optimization problem with the help of generalized invex functions. Further, we study saddle point criteria of a Lagrange function defined for a fractional interval-valued optimization problem. [ABSTRACT FROM AUTHOR]

This article focuses on optimality conditions for a robust fractional interval-valued optimization problem with uncertain inequality constraints (RNFIVP) based on convexificators. Using the tools of convexity, an example of sufficient optimality conditions is demonstrated. Robust parametric duality for (RNFIVP) is formulated and utilizing the concept of convexity, usual duality results between the primal and dual problems are investigated. Further, the equivalence between the saddle point criteria of a Lagrangian type function and a robust LU L U $ \mathcal{L}\mathcal{U}$ -optimal solution for (RNFIVP) with convexity is also examined. [ABSTRACT FROM AUTHOR]

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This paper is devoted to zero-sum piecewise deterministic Markov games with Borel state and action spaces, where the expected infinite-horizon discounted payoff criterion is considered. Both the transition rate and payoff rate are allowed to be unbounded. The policies of the two players are history-dependent, and the controls continuously act on the transition rate and the payoff rate. Under suitable conditions, Dynkin's formula and the comparison theorem are developed in our setup, via which the game is shown to have the value function as the unique solution to the associated Shapley equation. By the Shapley equation in the form of a differential equation, we establish the existence of a saddle point with a very simple form, which only depends on the current state and can be applied at any time. A potential algorithm for computing saddle points is proposed. [ABSTRACT FROM AUTHOR]

We consider a predator‐prey model with dissimilar functional and numerical responses that induce an Allee effect. There is a time lag between consumption and digestion of prey biomass by predator. Hence, a time delay has been incorporated in the numerical response function. The system consists of two interior equilibria. Taking time delay as the bifurcation parameter, four different dynamic behaviors appear, viz., (R1) system undergoes no change in its stability for all time delay, (R2) system undergoes stability change, (R3) system undergoes stability switching, and (R4) system undergoes instability switching. Here, finding four distinct dynamics in a single population model with only one delay is a novelty in this contribution. This variation in dynamics emerges due to asymmetricity in functional and numerical responses. All the relevant theorems in establishing stability are provided, and these are verified numerically. We analytically prove that if an interior equilibrium is a saddle point in absence of time delay, then the equilibrium cannot be stabilized by varying the time delay. It is popularly believed that existence of two distinct pair of purely imaginary roots of the characteristic function leads to stability switching. However, we provide examples where the system remains unstable, stability changes, and instability switching occurs. This is another new and interesting observation in our work. The numerical examples are furnished with phase portraits, time series plots, bifurcation diagrams, and eigenvalues evaluation with delay, for better understanding. Our model with a single delay exhibits variety of dynamics, which were not explored before. [ABSTRACT FROM AUTHOR]

The inverse velocity (INV) method is widely used for predicting the slope failure time. When applying the INV method, the inverse velocity can be assumed to be a linear and non-linear function of time, respectively, which are called linear and non-linear INV methods in this paper, respectively. Very few guidance is available in the literatures on the use of the two types of INV methods. In this paper, the performances of the linear and non-linear INV methods are assessed using a landslide database with 55 case histories. It is found that, two types of pitfalls may be encountered when applying the non-linear INV method, i.e. the saddle point and the ill-conditioned Hessian matrix. For the landslides examined in this paper, the linear INV method is free from the two pitfalls. When these pitfalls are encountered, the failure time predicted based on the non-linear INV methods may be significantly different from the actual slope failure time. For the landslides examined in this paper, the linear INV method is not only more stable, but also more accurate than the non-linear INV method. It is suggested that the linear INV method should be preferred over the non-linear INV method in future applications. [ABSTRACT FROM AUTHOR]

Complex nonlinear optimal control problems can have more than a single local solution due to the lack of convexity, and widely-used gradient-based approaches for optimal control have a risk of producing an inferior local solution. Although global optimization methods for optimal control may overcome this difficulty, such existing methods suffer from high computational complexities. This paper presents a novel numerical algorithm for multi-modal optimal control problems based on a saddle-point search method. Starting from an already-found local solution, a saddle-point search algorithm finds a surrounding saddle point of the constrained optimization problem, from which a new local solution is obtained by applying a gradient-based optimizer. By repeating this procedure, multiple locally optimal solutions are explored exhaustively and systematically. Although the developed method is not rigorously guaranteed to produce the global solution, it is applicable to large-scale general optimal control problems with equality and inequality constraints, which makes it a practically beneficial technique. After numerical experiments with an example problem, the ascent trajectory optimization problem of a winged rocket for maximizing apogee altitude is solved by using the proposed method.

Abstract In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: a ( u , v − u ) + b ( v − u , λ ) + j ( v ) − j ( u ) + J ( u , v ) − J ( u , u ) ≥ ( f , v − u ) X , b ( u , μ − λ ) − ψ ( μ ) + ψ ( λ ) ≤ 0 , $$\begin{aligned} &a(u,v-u)+b(v-u,\lambda )+j(v)-j(u)+J(u,v)-J(u,u)\geq (f,v-u)_{X}, \\ &b(u,\mu -\lambda )-\psi (\mu )+\psi (\lambda )\leq 0, \end{aligned}$$ ( v ∈ K ⊆ X $v\in K\subseteq X$ , μ ∈ Λ ⊂ Y $\mu \in \Lambda \subset Y$ ), where ( X , ( ⋅ , ⋅ ) X ) $(X,(\cdot,\cdot )_{X})$ and ( Y , ( ⋅ , ⋅ ) Y ) $(Y,(\cdot,\cdot )_{Y})$ are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting ψ ( ⋅ ) = ϵ ψ ¯ ( ⋅ ) $\psi (\cdot )=\epsilon \bar{\psi}(\cdot )$ ( ϵ > 0 ) $(\epsilon >0)$ . Then, we deliver a convergence result for ϵ → 0 $\epsilon \to 0$ , the case ψ ≡ 0 $\psi \equiv 0$ appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones.

We formulate a variational problem for harmonic waves in a viscoelastic body with short-term memory generated by distributed loads of a given frequency. The conditions of its well-posedness and equivalence to the saddle-point problem for the corresponding Lagrangian are established. We construct an h -adaptive scheme of the finite-element method for the solution of the formulated problem with an a posteriori error estimator determined element by element and a criterion for the local improvement of Delaunay triangulations used for the evaluation of approximations with prescribed accuracy. The efficiency of the proposed methods is illustrated by an example of numerical investigation of the resonance frequencies of a square plate containing a square hole. [ABSTRACT FROM AUTHOR]

Finding the saddle point of a matrix game is a classical problem that arises in various fields, e.g., economics, computer science, and engineering. The standard problem-solving methods consist of formulating the problem as a linear program (LP). However, this approach seems less efficient when many instances need to be solved. In this paper, we propose a Convolutional Neural Network based approach, which is able to predict both the strategy profile (x , y) and the optimal value v of the game. We call this approach Matrix Game-Conventional Neural Network or MG-CNN for short. Thanks to a global pooling technique, MG-CNN can solve matrix games with different shapes. We propose a specialized algorithm to train MG-CNN, which includes both data generation and model training. Our numerical experiments show that MG-CNN outperforms standard LP solvers in terms of computational CPU time and provides a high-quality prediction. [ABSTRACT FROM AUTHOR]

In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: a (u , v − u) + b (v − u , λ) + j (v) − j (u) + J (u , v) − J (u , u) ≥ (f , v − u) X , b (u , μ − λ) − ψ (μ) + ψ (λ) ≤ 0 , (v ∈ K ⊆ X , μ ∈ Λ ⊂ Y ), where (X , (⋅ , ⋅) X) and (Y , (⋅ , ⋅) Y) are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting ψ (⋅) = ϵ ψ ¯ (⋅) (ϵ > 0) . Then, we deliver a convergence result for ϵ → 0 , the case ψ ≡ 0 appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones. [ABSTRACT FROM AUTHOR]

The saddle point (SP) calculation is a grand challenge for computationally intensive energy function in computational chemistry area, where the saddle point may represent the transition state. The traditional methods need to evaluate the gradients of the energy function at a very large number of locations. To reduce the number of expensive computations of the true gradients, we propose an active learning framework consisting of a statistical surrogate model, Gaussian process regression (GPR) for the energy function, and a single-walker dynamics method, gentle accent dynamics (GAD), for the saddle-type transition states. SP is detected by the GAD applied to the GPR surrogate for the gradient vector and the Hessian matrix. Our key ingredient for efficiency improvements is an active learning method which sequentially designs the most informative locations and takes evaluations of the original model at these locations to train GPR. We formulate this active learning task as the optimal experimental design problem and propose a very efficient sample-based sub-optimal criterion to construct the optimal locations. We show that the new method significantly decreases the required number of energy or force evaluations of the original model. [ABSTRACT FROM AUTHOR]