Advanced Tensor Solution to the Problem of Inter-Domain Routing with Normalized Quality of Service

The advanced tensor solution to the problem of inter-domain routing with normalized Quality of Service under hierarchical coordination in a communication network is proposed in the paper. The novelty of the method based on the tensor model lies in the more flexible load balancing over the network due to the presence of requirements to average end-to-end delay of packets. The framework of the method comprises a decomposed flow-based routing model that includes the inter-domain routing interaction conditions and ensures the normalized Quality of Service derived from the tensor model. Considering the mentioned above, the advanced inter-domain Quality of Service routing task was formulated in the optimization form, using the quadratic optimality criterion. The conducted analysis of the numerical research results confirmed the efficiency and adequacy of the proposed method when the desired solutions were obtained during the finite number of iterations under a provision of the normalized Quality of Service. It should be noted that the reduced number of such iterations during the operation of the method will decrease the amount of service traffic transmitted over the network needed for obtaining the final solution in the process of inter-domain routing with normalized Quality of Service.

Algebraic Solution of Tropical Polynomial Optimization Problems

We consider constrained optimization problems defined in the tropical algebra setting on a linearly ordered, algebraically complete (radicable) idempotent semifield (a semiring with idempotent addition and invertible multiplication). The problems are to minimize the objective functions given by tropical analogues of multivariate Puiseux polynomials, subject to box constraints on the variables. A technique for variable elimination is presented that converts the original optimization problem to a new one in which one variable is removed and the box constraint for this variable is modified. The novel approach may be thought of as an extension of the Fourier–Motzkin elimination method for systems of linear inequalities in ordered fields to the issue of polynomial optimization in ordered tropical semifields. We use this technique to develop a procedure to solve the problem in a finite number of iterations. The procedure includes two phases: backward elimination and forward substitution of variables. We describe the main steps of the procedure, discuss its computational complexity and present numerical examples.

On Exchange Methods for Nonlinear Semi-Infinite Programs

A new exchange method is presented for semi-infinite optimization problems with polyhedron constraints. The basic idea is to use an active set strategy as exchange rule to construct an approximate problem with finitely many constraints at each iteration. Under mild conditions, we prove that the proposed algorithm terminates in a finite number of iterations and guarantees that the solution of the resulting approximate problem at final iteration converges to the solution of the original problem within arbitrarily given tolerance. Numerical results indicate that the proposed algorithm is efficient and promising.

Effective Distribution of Tasks in Multiprocessor and Multi-Computers Distributed Homogeneous Systems

Nowadays, a promising is the direction associated with the use of a large number of processors to solve the resource-intensive tasks. The enormous potential of multiprocessor and multicomputer systems can be fully revealed only when we apply effective methods for organizing the distribution of tasks between processors or computers. However, the problem of efficient distribution of tasks between processors and computers in similar computing systems remains relevant. Two key factors are critical and have an impact on system performance. This is load uniformity and interprocessor or intercomputer interactions. These conflicting factors must be taken into account simultaneously in the distribution of tasks in multiprocessor computing systems. A uniform loading plays a key role in achieving high parallel efficiency, especially in systems with a large number of processors or computers. Efficiency means not only the ability to obtain the result of computations in a finite number of iterations with the necessary accuracy, but also to obtain the result in the shortest possible time. The number of tasks intended for execution on each processor or each computer should be determined so that the execution time is minimal. This study offers a technique that takes into account the workload of computers and intercomputer interactions, and allows one to minimize the execution time of tasks. The technique proposed by the authors allows the comparison of different architectures of computers and computing modules. In this case, a parameter is used that characterizes the behavior of various models with a fixed number of computers, as well as a parameter that is necessary to compare the effectiveness of each computer architecture or computing module when a different number of computers are used. The number of computers can be variable at a fixed workload. The mathematical implementation of this method is based on the problem solution of the mathematical optimization or feasibility.

Approximation of null controls for semilinear heat equations using a least-squares approach

The null distributed controllability of the semilinear heat equation $\partial_t y-\Delta y + g(y)=f \,1_{\omega}$ assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized wave equation. Assuming that $g^\prime$ is bounded and uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in (0,1]$, we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton methods: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.

On the Locally Polynomial Complexity of the Projection-Gradient Method for Solving Piecewise Quadratic Optimisation Problems

This paper proposes a method for solving optimisation problems involving piecewise quadratic functions. The method provides a solution in a finite number of iterations, and the computational complexity of the proposed method is locally polynomial of the problem dimension, i.e., if the initial point belongs to the sufficiently small neighbourhood of the solution set. Proposed method could be applied for solving large systems of linear inequalities.

Direct Method for Solving Bilinear Programming Problem

The bilinear programming problem is considered, where a column, which corresponds to one of the variables, is not fixed but can be chosen from a convex set. This problem is known as the Dantzig – Wolfe problem. Earlier, a modified support method was proposed to solve the problem, using the decomposition of the problem constraints of the Dantzig – Wolfe method. The author of the paper has developed a direct exact method for solving the formulated problem. The method is based on the idea of the solving a linear programming problem with generalized direct constraints and a general concept of an adaptive solution method. The notions of support, support plan, optimal and suboptimal (e-optimal) plan are introduced which is a given approximation of the objective function to the optimal plan of the problem. Criteria for optimality and suboptimality of the support plan have been formulated and have been proved in the paper. The search for the optimal solution is based on the idea of maximizing the increment of the objective function. This approach allows more fully to take into account the main target and structure of the problem. Improving a support plan consists of two parts: replacing the plan and replacing the support. To find a suitable direction, a special derived problem is solved while taking into account the main constraints of the problem. The replacement of the support is based on the search for the optimal plan of the dual problem. The method leads to an optimal solution to the problem in a finite number of iterations (in the case of a non-degenerate value).

Nilpotent endomorphisms of expansive group actions

We consider expansive group actions on a compact metric space containing a special fixed point denoted by [Formula: see text], and endomorphisms of such systems whose forward trajectories are attracted toward [Formula: see text]. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to [Formula: see text] in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in particular shows that the class includes all residually finite solvable groups and all groups of polynomial growth. For expansive actions of the group [Formula: see text], we show that a very weak gluing property suffices for nil-rigidity. For [Formula: see text]-subshifts of finite type, we show that the block-gluing property suffices. The study of nil-rigidity is motivated by two aspects of the theory of cellular automata and symbolic dynamics: It can be seen as a finiteness property for groups, which is representative of the theory of cellular automata on groups. Nilpotency also plays a prominent role in the theory of cellular automata as dynamical systems. As a technical tool of possible independent interest, the proof involves the construction of tiered dynamical systems where several groups act on nested subsets of the original space.

Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

We consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

A transformation between stationary point vortex equilibria

A new transformation between stationary point vortex equilibria in the unbounded plane is presented. Given a point vortex equilibrium involving only vortices with negative circulation normalized to −1 and vortices with positive circulations that are either integers or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant. When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations, each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko (Loutsenko 2004 J. Phys. A: Math. Gen. 37 , 1309–1321 (doi:10.1088/0305-4470/37/4/017)). For the latter polynomials, the existence of such a transformation appears to be new. The new transformation, therefore, unifies a wide range of disparate results in the literature on point vortex equilibria.