Set Of Stationary Points Research Articles

A Tractable Multi-Leader Multi-Follower Peak-Load-Pricing Model with Strategic Interaction

While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. Finally, we apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system.

Numerical construction of spherical t-designs by Barzilai-Borwein method

A point set XN on the unit sphere is a spherical t-design is equivalent to the nonnegative quantity AN,t+1 vanished. We show that if XN is a stationary point set of AN,t+1 and the minimal singular value of basis matrix is positive, then XN is a spherical t-design. Moreover, the numerical construction of spherical t-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical t-designs with N=(t+2)2 points for t+1 up to 127.

Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition

The Levenberg–Marquardt method is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares problems. In this paper, we consider local convergence properties of the method, when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full rank of the Jacobian in a neighborhood of a stationary point. Differently from the zero-residue case, the choice of the Levenberg–Marquardt parameter is shown to be dictated by (i) the behavior of the rank of the Jacobian and (ii) a combined measure of nonlinearity and residue size in a neighborhood of the set of (possibly non-isolated) stationary points of the sum of squares function.

Nonconvex Proximal Incremental Aggregated Gradient Method with Linear Convergence

In this paper, we study the proximal incremental aggregated gradient(PIAG) algorithm for minimizing the sum of L-smooth nonconvex component functions and a proper closed convex function. By exploiting the L-smooth property and with the help of an error bound condition, we can show that the PIAG method still enjoys some nice linear convergence properties even for nonconvex minimization. To illustrate this, we first demonstrate that the generated sequence globally converges to the stationary point set. Then, there exists a threshold such that the objective function value sequence and the iterate point sequence are R-linearly convergent when the stepsize is chosen below this threshold.

О ГАМИЛЬТОНОВОЙ РЕДУКЦИИ УРАВНЕНИЙ АССОЦИАТИВНОСТИ В СЛУЧАЕ ЧЕТЫРЕХ ПРИМАРНЫХ ПОЛЕЙ

The associativity equations arose in the papers of Witten (Witten, 1990) and Dijkgraaf, Verlinde, (Dijkgraaf et al., 1991) on two-dimensional topological field theories and subsequently they became to play a key role in many other important domains of mathematics and mathematical physics: in quantum cohomology, Gromov–Witten invariants, enumerative geometry, theory of submanifolds and so on. In Mokhov’s papers (Mokhov, 1984), (Mokhov, 1987) a general fundamental principle stating a canonical Hamiltonian property for the restriction of an arbitrary flow on the set of stationary points of its nondegenerate integral was proposed and proved. In this paper the Hamiltonians of the reductions of the associativity equations with antidiagonal matrix ηij in the case of four primary fields according to Mokhov`s construction is found in an explicit form.

ИНТЕГРИРУЕМЫЕ ПО ЛИУВИЛЛЮ РЕДУКЦИИ УРАВНЕНИЙ АССОЦИАТИВНОСТИ НА МНОЖЕСТВО СТАЦИОНАРНЫХ ТОЧЕК ИНТЕГРАЛА В СЛУЧАЕ ТРЕХ ПРИМАРНЫХ ПОЛЕЙ

In this work, in the case of three primary fields, a reduction of the associativity equations (the Witten–Dijkgraaf–Verlinde–Verlinde system, see (Witten, 1990, Dijkgraaf et al., 1991, Dubrovin, 1994) with antidiagonal matrix ηij on the set of stationary points of a nondegenerate integral quadratic with respect to the first-order partial derivatives is constructed in an explicit form and its Liouville integrability is proved. In Mokhov’s paper (Mokhov, 1995, Mokhov, 1998), these associativity equations were presented in the form of an integrable nondiagonalizable system of hydrodynamic type. In the papers (Ferapontov, Mokhov, 1996, Ferapontov et al., 1997, Mokhov, 1998), a bi-Hamiltonian representation for these equations and a nondegenerate integral quadratic with respect to the first-order partial derivatives were found. Using Mokhov’s construction on canonical Hamiltonian property of an arbitrary evolutionary system on the set of stationary points of its nondegenerate integral of the papers (Mokhov, 1984, Mokhov, 1987), we construct explicitly the reduction for the integral quadratic with respect to the first-order partial derivatives, found explicitly the Hamiltonian of the corresponding canonical Hamiltonian system. We also found three functionally-independent integrals in involution with respect to the canonical Poisson bracket on the phase space for the constructed reduction of the associativity equations and thus proved the Liouville integrability of this reduction. This work is supported by the Russian Science Foundation under grant No. 18-11-00316.

Liouville integrability of the reduction of the associativity equations on the set of stationary points of an integral in the case of three primary fields

Liouville integrability of the reduction of the associativity equations on the set of stationary points of an integral in the case of three primary fields

On the Numbers of Connected Components in the Solution Sets of Polynomial Vector Variational Inequalities

In this paper, we establish explicit upper bounds for the number of connected components in the proper Pareto solution sets and the weak Pareto solution sets of polynomial vector variational inequalities. Consequently, upper bounds for the numbers of connected components in the stationary point sets, the proper stationary point sets, and the weak Pareto solution sets of polynomial vector optimization problems are obtained.

Closed‐loop supply chain network static equilibrium and dynamics under pollution permits system

A closed‐loop supply chain network including multi‐manufacturing/remanufacturing firms and multi‐demand markets is considered. We present equilibrium conditions for decision‐makers in the network and develop a static model with marketable pollution permits and noncompliant behavior, using the variational inequality theory. We provide qualitative analysis of the model. Then, using the projected dynamical system, we propose a dynamic model. The results show that the set of stationary points of dynamic model coincides with the set of solutions of static model. Finally, numerical examples are proposed. The equilibrium results including production output of new products, the forward and reverse product flows, the prices of product, the prices of license, the number of licenses, and the possible noncompliant overflows and underflows of the waste are obtained. We find that increase in the unit penalty cost contributes to decrease in waste emission from firms in the network, and when the unit penalty cost increase to a certain level, they no longer choose excess emission of waste. © 2018 American Institute of Chemical Engineers Environ Prog, 38:e13021, 2019

Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability

By applying some theorems of Levy and Mordukhovich (Math Program 99: 311--327, 2004) and other related results, we estimate the Fr\'echet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From the obtained formulas we derive necessary and sufficient conditions for the local Lipschitz-like property of the stationary point set map. This leads us to new insights into the preceding deep investigations of Levy and Mordukhovich in the above-cited paper and of Qui (J Optim Theory Appl 161: 398--429, 2014; J Glob Optim 65: 615--635, 2016).

Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 2: Robinson Stability

In Part 1 of this paper, we have estimated the Fr\'echet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given.

A unified framework for sparse non-negative least squares using multiplicative updates and the non-negative matrix factorization problem

We study the sparse non-negative least squares (S-NNLS) problem. S-NNLS occurs naturally in a wide variety of applications where an unknown, non-negative quantity must be recovered from linear measurements. We present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior on the sparse codes. We show that the proposed framework encompasses a large class of S-NNLS algorithms and provide a computationally efficient inference procedure based on multiplicative update rules. Such update rules are convenient for solving large sets of S-NNLS problems simultaneously, which is required in contexts like sparse non-negative matrix factorization (S-NMF). We provide theoretical justification for the proposed approach by showing that the local minima of the objective function being optimized are sparse and the S-NNLS algorithms presented are guaranteed to converge to a set of stationary points of the objective function. We then extend our framework to S-NMF, showing that our framework leads to many well known S-NMF algorithms under specific choices of prior and providing a guarantee that a popular subclass of the proposed algorithms converges to a set of stationary points of the objective function. Finally, we study the performance of the proposed approaches on synthetic and real-world data.

A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized Nonnegative Matrix Factorization

The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the nonconvex NMF optimization problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing.

Online Nonnegative Matrix Factorization With Outliers

We propose a unified and systematic framework for performing online nonnegative matrix factorization in the presence of outliers. Our framework is particularly suited to large-scale data. We propose two solvers based on projected gradient descent and the alternating direction method of multipliers. We prove that the sequence of objective values converges almost surely by appealing to the quasi-martingale convergence theorem. We also show the sequence of learned dictionaries converges to the set of stationary points of the expected loss function almost surely. In addition, we extend our basic problem formulation to various settings with different constraints and regularizers. We also adapt the solvers and analyses to each setting. We perform extensive experiments on both synthetic and real datasets. These experiments demonstrate the computational efficiency and efficacy of our algorithms on tasks such as (parts-based) basis learning, image denoising, shadow removal and foreground-background separation.

Online Categorical Subspace Learning for Sketching Big Data with Misses

With the scale of data growing every day, reducing the dimensionality (a.k.a. sketching) of high-dimensional data has emerged as a task of paramount importance. Relevant issues to address in this context include the sheer volume of data that may consist of categorical samples, the typically streaming format of acquisition, and the possibly missing entries. To cope with these challenges, the present paper develops a novel categorical subspace learning approach to unravel the latent structure for three prominent categorical (bilinear) models, namely, Probit, Tobit, and Logit. The deterministic Probit and Tobit models treat data as quantized values of an analog-valued process lying in a low-dimensional subspace, while the probabilistic Logit model relies on low dimensionality of the data log-likelihood ratios. Leveraging the low intrinsic dimensionality of the sought models, a rank regularized maximum-likelihood estimator is devised, which is then solved recursively via alternating majorization-minimization to sketch high-dimensional categorical data `on the fly.' The resultant procedure alternates between sketching the new incomplete datum and refining the latent subspace, leading to lightweight first-order algorithms with highly parallelizable tasks per iteration. As an extra degree of freedom, the quantization thresholds are also learned jointly along with the subspace to enhance the predictive power of the sought models. Performance of the subspace iterates is analyzed for both infinite and finite data streams, where for the former asymptotic convergence to the stationary point set of the batch estimator is established, while for the latter sublinear regret bounds are derived for the empirical cost. Simulated tests with both synthetic and real-world datasets corroborate the merits of the novel schemes for real-time movie recommendation and chess-game classification.

Online Sparsifying Transform Learning—Part II: Convergence Analysis

Sparsity-based techniques have been widely popular in signal processing applications such as compression, denoising, and compressed sensing. Recently, the learning of sparsifying transforms for data has received interest. The advantage of the transform model is that it enables cheap and exact computations. In Part I of this work, efficient methods for online learning of square sparsifying transforms were introduced and investigated (by numerical experiments). The online schemes process signals sequentially, and can be especially useful when dealing with big data, and for real-time, or limited latency signal processing applications. In this paper, we prove that although the associated optimization problems are non-convex, the online transform learning algorithms are guaranteed to converge to the set of stationary points of the learning problem. The guarantee relies on a few simple assumptions. In practice, the algorithms work well, as demonstrated by examples of applications to representing and denoising signals.

A relaxed SCA approach for calibrating low rank correlation matrix problem

In this paper, we consider a class of rank constrained correlation matrix calibrating problems with simple upper and lower bounds. This problem can be reformulated into a DC (difference of convex) constrained problem, thus sequential convex approximation (SCA) type approaches can be considered. However, classical SCA approach cannot be directly applied since the constraint qualification needed in convergence theorem does not hold for this problem. This motivates us to develop a relaxed SCA approach. We prove that for relaxed DC problems, the sets of their stationary points converge to the set of stationary points of the original problem as the relaxation parameter approaching to zero. For each relaxed DC problem, we apply the SCA approach to generate a sequence of convex subproblems. We show that all cluster points of optimal solutions of these subproblems are stationary points of the relaxed DC problem. Numerical results verify the efficiency of our approach.

Hyperspaces and the S-equivariant Complete Invariance Property

In this paper it is investigated as to when a nonempty invariant closed subset A of a -space X containing the set of stationary points (S) can be the fixed point set of an equivariant continuous selfmap on X and such space X is said to possess the S-equivariant complete invariance property (S-ECIP). It is also shown that if X is a metric space and acts on by the action , where p, and , then the hyperspace of all nonempty compact subsets of has the S-ECIP.

Generic Half-Quadratic Optimization for Image Reconstruction

We study the global and local convergence of a generic half-quadratic optimization algorithm inspired from the dual energy formulation of Geman and Reynolds [IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), pp. 367--383]. The target application is the minimization of $C^{1}$ convex and nonconvex objective functionals arising in regularized image reconstruction. Our global convergence proofs are based on a monotone convergence theorem of Meyer [J. Comput. System Sci., 12 (1976), pp. 108--121]. Compared to existing results, ours extend to a larger class of objectives and apply under weaker conditions; in particular, we cover the case where the set of stationary points is not discrete. Our local convergence results use a majorization-minimization interpretation to derive an insightful characterization of the basins of attraction; this new perspective grounds a formal description of the intuitive water-flooding analogy. We conclude with image restoration experiments to illustrate the efficiency of the algo...

Relaxed conditions for convergence of batch BPAP for feedforward neural networks

In this paper, the deterministic convergence of the batch back-propagation algorithm with penalty (BPAP) is proved under certain relaxed conditions for the activation function, the learning rate and the stationary point set of the error function. Both weak and strong convergence results are established. The boundedness of the weights in the training procedure is also proved in a simple and clear way. As a result, the usual requirements on the boundedness of the weights to guarantee the convergence are removed. Simulation results for an approximation problem are presented to support our theoretical findings.