Oleg Burdakov

Positioning Unmanned Aerial Vehicles As Communication Relays for Surveillance Tasks

When unmanned aerial vehicles (UAVs) are used to survey distant targets, it is important to transmit sensor information back to a base station. As this communication often requires high uninterrupted bandwidth, the surveying UAV often needs afree line-of-sight to the base station, which can be problematic in urban or mountainous areas. Communication ranges may also belimited, especially for smaller UAVs. Though both problems can be solved through the use of relay chains consisting of one or more intermediate relay UAVs, this leads to a new problem: Where should relays be placed for optimum performance? We present two new algorithms capable of generating such relay chains, one being a dual ascent algorithm and the other a modification of the Bellman-Ford algorithm. As the priorities between the numberof hops in the relay chain and the cost of the chain may vary, wecalculate chains of different lengths and costs and let the ground operator choose between them. Several different formulations for edge costs are presented. In our test cases, both algorithms are substantially faster than an optimized version of the original Bellman-Ford algorithm, which is used for comparison.

Relay Positioning for Unmanned Aerial Vehicle Surveillance

When unmanned aerial vehicles (UAVs) are used for surveillance, information must often be transmitted to a base station in real time. However, limited communication ranges and the common requirement of free line of sight may make direct transmissions from distant targets impossible. This problem can be solved using relay chains consisting of one or more intermediate relay UAVs. This leads to the problem of positioning such relays given known obstacles, while taking into account a possibly mission-specific quality measure. The maximum quality of a chain may depend strongly on the number of UAVs allocated. Therefore, it is desirable to either generate a chain of maximum quality given the available UAVs or allow a choice from a spectrum of Pareto-optimal chains corresponding to different trade-offs between the number of UAVs used and the resulting quality. In this article, we define several problem variations in a continuous three-dimensional setting. We show how sets of Pareto-optimal chains can be generated using graph search and present a new label-correcting algorithm generating such chains significantly more efficiently than the best-known algorithms in the literature. Finally, we present a new dual ascent algorithm with better performance for certain tasks and situations.

Generating UAV communication networks for monitoring and surveillance

An important use of unmanned aerial vehicles is surveillance of distant targets, where sensor information must quickly be transmitted back to a base station. In many cases, high uninterrupted bandwidth requires line-of-sight between sender and transmitter to minimize quality degradation. Communication range is typically limited, especially when smaller UAVs are used. Both problems can be solved by creating relay chains for surveillance of a single target, and relay trees for simultaneous surveillance of multiple targets. In this paper, we show how such chains and trees can be calculated. For relay chains we create a set of chains offering different trade-offs between the number of UAVs in the chain and the chains cost. We also show new results on how relay trees can be quickly calculated and then incrementally improved if necessary. Encouraging empirical results for improvement of relay trees are presented.

Optimal placement of UV-based communications relay nodes

We consider a constrained optimization problem with mixed integer and real variables. It models optimal placement of communications relay nodes in the presence of obstacles. This problem is widely encountered, for instance, in robotics, where it is required to survey some target located in one point and convey the gathered information back to a base station located in another point. One or more unmanned aerial or ground vehicles (UAVs or UGVs) can be used for this purpose as communications relays. The decision variables are the number of unmanned vehicles (UVs) and the UV positions. The objective function is assumed to access the placement quality. We suggest one instance of such a function which is more suitable for accessing UAV placement. The constraints are determined by, firstly, a free line of sight requirement for every consecutive pair in the chain and, secondly, a limited communication range. Because of these requirements, our constrained optimization problem is a difficult multi-extremal problem for any fixed number of UVs. Moreover, the feasible set of real variables is typically disjoint. We present an approach that allows us to efficiently find a practically acceptable approximation to a global minimum in the problem of optimal placement of communications relay nodes. It is based on a spatial discretization with a subsequent reduction to a shortest path problem. The case of a restricted number of available UVs is also considered here. We introduce two label correcting algorithms which are able to take advantage of using some peculiarities of the resulting restricted shortest path problem. The algorithms produce a Pareto solution to the two-objective problem of minimizing the path cost and the number of hops. We justify their correctness. The presented results of numerical 3D experiments show that our algorithms are superior to the conventional Bellman-Ford algorithm tailored to solving this problem.

Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

Optimization problems with cardinality constraints are very difficult mathematical programs which are typically solved by global techniques from discrete optimization. Here we introduce a mixed-integer formulation whose standard relaxation still has the same solutions (in the sense of global minima) as the underlying cardinality-constrained problem; the relation between the local minima is also discussed in detail. Since our reformulation is a minimization problem in continuous variables, it allows us to apply ideas from that field to cardinality-constrained problems. Here, in particular, we therefore also derive suitable stationarity conditions and suggest an appropriate regularization method for the solution of optimization problems with cardinality constraints. This regularization method is shown to be globally convergent to a Mordukhovich-stationary point. Extensive numerical results are given to illustrate the behavior of this method.

An O(n2) algorithm for isotonic regression

We consider the problem of minimizing the distance from a given n-dimensional vector to a set defined by constraints of the form x i ≤ x j. Such constraints induce a partial order of the components x i, which can be illustrated by an acyclic directed graph. This problem is also known as the isotonic regression (IR) problem. IR has important applications in statistics, operations research and signal processing, with most of them characterized by a very large value of n. For such large-scale problems, it is of great practical importance to develop algorithms whose complexity does not rise with n too rapidly. The existing optimization-based algorithms and statistical IR algorithms have either too high computational complexity or too low accuracy of the approximation to the optimal solution they generate. We introduce a new IR algorithm, which can be viewed as a generalization of the Pool-Adjacent-Violator (PAV) algorithm from completely to partially ordered data. Our algorithm combines both low computational complexity O(n 2) and high accuracy. This allows us to obtain sufficiently accurate solutions to IR problems with thousands of observations.

A Limited-Memory Multipoint Symmetric Secant Method for Bound Constrained Optimization

A new algorithm for solving smooth large-scale minimization problems with bound constraints is introduced. The way of dealing with active constraints is similar to the one used in some recently introduced quadratic solvers. A limited-memory multipoint symmetric secant method for approximating the Hessian is presented. Positive-definiteness of the Hessian approximation is not enforced. A combination of trust-region and conjugate-gradient approaches is used to explore useful information. Global convergence is proved for a general model algorithm. Results of numerical experiments are presented.

Inverse Monte Carlo in a multilayered tissue model: merging diffuse reflectance spectroscopy and laser Doppler flowmetry.

Abstract. The tissue fraction of red blood cells (RBCs) and their oxygenation and speed-resolved perfusion are estimated in absolute units by combining diffuse reflectance spectroscopy (DRS) and laser Doppler flowmetry (LDF). The DRS spectra (450 to 850 nm) are assessed at two source–detector separations (0.4 and 1.2 mm), allowing for a relative calibration routine, whereas LDF spectra are assessed at 1.2 mm in the same fiber-optic probe. Data are analyzed using nonlinear optimization in an inverse Monte Carlo technique by applying an adaptive multilayered tissue model based on geometrical, scattering, and absorbing properties, as well as RBC flow-speed information. Simulations of 250 tissue-like models including up to 2000 individual blood vessels were used to evaluate the method. The absolute root mean square (RMS) deviation between estimated and true oxygenation was 4.1 percentage units, whereas the relative RMS deviations for the RBC tissue fraction and perfusion were 19% and 23%, respectively. Examples of in vivo measurements on forearm and foot during common provocations are presented. The method offers several advantages such as simultaneous quantification of RBC tissue fraction and oxygenation and perfusion from the same, predictable, sampling volume. The perfusion estimate is speed resolved, absolute (% RBC×mm/s), and more accurate due to the combination with DRS.

Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions

We suggest here a least-change correction to available finite element (FE) solution. This postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the FE solution. Although our approach is presented for FEs, it admits natural extension to other numerical schemes, such as finite differences and finite volumes. For the postprocessing, a priori information about the monotonicity is assumed to be available, either for the whole domain or for a subdomain where the lost monotonicity is to be recovered. The obvious requirement is that such information is to be obtained without involving the exact solution, e.g. from expected symmetries of this solution. The postprocessing is based on solving a monotonic regression problem with some extra constraints. One of them is a linear equality-type constraint that models the conservativity requirement. The other ones are box-type constraints, and they originate from the discrete maximum principle. The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessed FE solution preserves the accuracy of the discrete FE approximation. We introduce an algorithm for solving the postprocessing problem. It can be viewed as a dual ascent method based on the Lagrangian relaxation of the equality constraint. We justify theoretically its correctness. Its efficiency is demonstrated by the presented results of numerical experiments.

On a Reformulation of Mathematical Programs with Cardinality Constraints

Mathematical programs with cardinality constraints are optimization problems with an additional constraint which requires the solution to be sparse in the sense that the number of nonzero elements, i.e. the cardinality, is bounded by a given constant. Such programs can be reformulated as a mixed-integer ones in which the sparsity is modeled with the use of complementarity-type constraints. It is shown that the standard relaxation of the integrality leads to a nonlinear optimization program of the striking property that its solutions (global minimizers) are the same as the solutions of the original program with cardinality constraints. Since the number of local minimizers of the relaxed program is typically larger than the number of local minimizers of the cardinality-constrained problem, the relationship between the local minimizers is also discussed in detail. Furthermore, we show under which assumptions the standard KKT conditions are necessary optimality conditions for the relaxed program. The main result obtained for such conditions is significantly different from the existing optimality conditions that are known for the somewhat related class of mathematical programs with complementarity constraints.