Real-time Bidding for Time Constrained Impression Contracts in First and Second Price Auctions - Theory and Algorithms

We study the optimal bids and allocations in a real-time auction for heterogeneous items subject to the requirement that specified collections of items of given types be acquired within given time constraints. The problem is cast as a continuous time optimization problem that can, under certain weak assumptions, be reduced to a convex optimization problem. Focusing on the standard first and second price auctions, we first show, using convex duality, that the optimal (infinite dimensional) bidding policy can be represented by a single finite vector of so-called ''pseudo-bids''. Using this result we are able to show that the optimal solution in the second price case turns out to be a very simple piecewise constant function of time. This contrasts with the first price case that is more complicated. Despite the fact that the optimal solution for the first price auction is genuinely dynamic, we show that there remains a close connection between the two cases and that, empirically, there is almost no difference between optimal behavior in either setting. This suggests that it is adequate to bid in a first price auction as if it were in fact second price. Finally, we detail methods for implementing our bidding policies in practice with further numerical simulations illustrating the performance.

A novel speed optimisation scheme for unmanned sailboats by sliding mode extremum seeking control without steady-state oscillation

This paper proposes a novel speed optimisation scheme for unmanned sailboats by sliding mode extremum seeking control (SMESC) without steady-state oscillation. In the sailing speed optimisation scheme, an initial sail angle of attack is first computed by a piecewise constant function in the feed forward block, which ensures a small deviation between sailing speed and the maximum speed. Second, the sailing speed approaches to maximum gradually by extremum search control (ESC) in the feedback block. In SMESC without steady-state oscillation, a switching law is designed to carry out the control transformation, so that the speed optimisation system carries out SMESC in the first convergence phase and ESC without steady-state oscillation in the second stability phase. This scheme combines the advantages of both control algorithms to maintain a faster convergence rate and to eliminate steady-state oscillation. Furthermore, the strict stability of the speed optimisation system is proved in this paper. Finally, we test a 12-m mathematical model of an unmanned sailboat in the simulation to demonstrate the effectiveness and robustness of this speed optimisation scheme.

The Kirchhoff Transformation and the Fick’s Second Law with Concentration-dependent Diffusion Coefficient

In this work it is considered the Fick’s second law in a context in which the diffusion coefficient depends on the concentration. It is employed the Kirchhoff transformation in order to simplify the mathematical structure of the Fick’s second law, giving rise to a more convenient description. In order to provide a general protocol, the diffusion coefficient will be assumed a piecewise constant function of the concentration. Exact formulas are presented for both the Kirchhoff transformation and its inverse, in such a way that there is no limit of accuracy. Some numerical examples are presented with the aid of a semi-implicit procedure associated with a finite difference approximation.

Gauge theories on compact toric manifolds

We compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

Optimization of Industrial Complex Power Supply by Means of Renewable Energy Sources

The usage of renewable energy sources and energy storage devices allows an enterprise to reduce its electricity supply costs. Significant savings can be achieved only in the case of a well-organized system of managing energy flows of the enterprise and electricity consumption. The puprpose of the study was to analyze the existing storage devices and renewable energy sources, and on the basis of the analysis, introduce an approach to organizing energy supply of the enterprise based on the usage of storage devices and renewable energy sources. The approach introduced implies the electricity purchase schedule curve optimization to minimize the total costs. The purchase schedule curve optimization for the case of energy storage usage is an ill-posed problem. Within the framework of solving the problem, the form of the curve was formalized to a piecewise constant function, which made it possible to solve it by applying multi-criteria optimization based on a modified method of directed random search. We analyzed a model case, for which an optimal purchase schedule curve was obtained using storage devices and solar panels, and the achievable economic effect from their use was graphed. The maximum value of operational costs, which makes the usage of energy storage and renewable energy source efficient, was found

Current-phase relation in layered superconducting structures of SIS’IS type

The dependence of the current density on the phase difference is investigated considering the layered superconducting structures of a SIS’IS type. To simplify the calculations, the quasiclassical equations for the Green’s functions in a t-representation are derived. An order parameter is considered as a piecewise constant function. To consider the general case, no restrictions on the dielectric layer transparency and the thickness of the intermediate layer are imposed. It was found that a new analytical expression for the current-phase relation can be used with the aim to obtain a number of previously known results arising in particular cases.

Predictive Geometric Models for Heat-Insulation Properties of Semi-Finished Fur and Down Products

This paper is devoted to geometric simulation of heat-insulation properties of fur and down products which are considered as multi-parameter and multi-component systems. We consider predictive models of heat resistance depended on physical characteristics of fur and pelt. There is a problem of construction co-ordinate geometric models on condition that the set of experimental data is limited. We solve the problem as a problem for static multi-component systems. The model is considered as a piecewise constant function in the space of input and output parameters. The paper proposes an algorithm of construction the clusters on the set of given experimental points. Moreover, we construct multidimensional convex covering on the set of the points. The covering is based on its two-dimensional projections. Results of the investigations allow us to substantiate producer’s choice of fur and down semi-finished products and its composition for manufacturing the product of special purpose. The method suggested in the paper may be one of geometric modulus of the software HYPER-DESCENT which has been developed formerly. Our geometric models together with software HYPER- DESCENT may be applied for simulation and prediction the properties of another multi- parametrical systems or technological processes of light industry.

Optimization of the electrical technology complex for high-performance induction heating lines

The paper discusses the design issues of a high-performance induction installation for heating ferromagnetic billets before processing on deforming equipment. Specific features of the technological process of heating ferromagnetic billets to plastic strain temperatures are noted. It is shown that in order to increase the efficiency of the process, the heating of large-sized preforms from ferromagnetic steel is advisable to be carried out in a two-frequency induction heater with two autonomous sections. A study of the heating process in a two-section heater of ferromagnetic billets was carried out taking into account the nonlinear dependence of the physical characteristics of the heated billets metal on the temperature changing during heating. The problem of minimizing the length of a two-section heater is formulated and solved taking into account energy and technological limitations. A condition for determining the optimal length of the first section is to achieve a temperature corresponding to the loss of magnetic properties in the layer equal to the depth of current penetration at the frequency of the second section power supply. The results of numerical calculation of the optimal parameters of a two-section heater are presented. It is shown that the power distribution algorithm along the length of a two-section heater is a piecewise constant function. The results of calculating the temperature distribution in the workpieces during heating are presented. The results of the study can be used to solve the problem of optimizing the structural and operational parameters of a multi-section heater.

The Component Weighted Median Absolute Deviations Problem

This paper considers the problem of robust modeling by using the well-known Least Absolute Deviation (LAD) regression. For that purpose, the approximation function is designed and analyzed, which is based on a certain component weight of the Weighted Median of Data. It is shown that the proposed approximation function is a piecewise constant function with finitely many pieces with respect to the model parameter. Thereby, an investigation of regions of constant values of the approximation function is conducted. It is established that the designed model based on the Component Weighted Median Absolute Deviations estimates a optimal model parameter on a finite set, which describes corresponding regions. Furthermore, the specified restriction of the approximation function is observed and analyzed, in order to examine the observed problem.

Stabilization of a multiconnected controlled continuous discrete system with non-overlapped decompositions with respect to part of variables

This paper considers a multi-connected controllable system with non-overlapping decompositions. Given that most of the control laws are implemented on digital controllers, the control of the system is implemented as a piecewise-constant function. Multiconnectivity of the system, in turn, makes it impossible to use centralized control. Every isolated subsystem must work stably, and intersystem connections can have a destabilizing effect. In this case, piecewise-constant control is constructed as two-level, i.e. in the form of a sum of local and global control. Local control stabilizes the equilibrium positions of individual linear subsystems. Global control acts on intersystem connections. Conditions are obtained under which local control stabilizes linear subsystems, and the equilibrium position of the original multi-connected system is asymptotically stable in part of variables.