# Data-Driven Risk-Averse Optimization

Risk-awareness is of fundamental importance in many application areas, such as ** Finance, Energy, Smart Grid / Cities, Transportation, Supply Chain Management, Network control, Resource Allocation, Telecommunications, Robotics, and Big Data**. Still, except for certain batch methods, design of data-driven algorithms for risk-averse optimization is very unexplored, especially in

**and**

*nonstationary***, or when**

*real-time settings***is an operational constraint.**

*computational efficiency*The current state of the art presents a high potential for the development of new methods for optimization of risk. To date, I have contributed in data-driven optimization of a new class of convex risk measures, termed *mean-semideviations*, strictly generalizing the classical central mean-upper-semideviation risk measure. I have introduced the * MESSAGEp algorithm*, a compositional subgradient procedure for iteratively solving convex mean-semideviation problems to optimality. The

**algorithm may be seen as a variation of the general purpose**

*MESSAGEp*

*T-SCGD**algorithm*of Yang, Wang & Fang ([YWF18]), originally analyzed under a generic setting. By exploiting problem structure, I have proposed a new, substantially more flexible set of assumptions, as compared to [YWF18], under which I have established pathwise convergence and convergence rates of the

**algorithm, extending and improving on the state of the art. I have also rigorously shown that the new framework strictly generalizes [YWF18], allowing for much less restrictive problem requirements and establishing the applicability of compositional stochastic optimization for a strictly wider, extended spectrum of convex mean-semideviation risk-averse problems.**

*MESSAGEp***Support:** DARPA Lagrange program, U.S. Navy / SPAWAR Systems Center Pacific under Contract No. N66001-18-C-4031.

**Selected Publications:**

- D. S. Kalogerias and W. B. Powell, “Recursive Optimization of Convex Risk Measures: Mean-Semideviation Models,” Working Paper, March 2018 (latest update: October 2018).

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# Spatially Controlled

Relay Communications

Distributed, networked communication systems are typically designed without considering how node positioning might affect Quality-of-Service (QoS). Network nodes are either assumed to be stationary or, if moving, their trajectories are independent of the respective communication task. In reality, though, channel information observed by each node is both spatially and temporally dependent. It is thus reasonable to ask if and how system performance can be improved by strategically controlling node positions, exploiting the spatiotemporal dependencies of the wireless channel.

I have developed a rigorous framework for exploiting node mobility in cooperative networks, in a fully stochastic setting. The basic paradigm considered is a single endpoint relay beamforming network, where relaying nodes are spatially controllable.

*sequentially executed*2-stage stochastic programming framework, which optimizes expected network QoS

*jointly*over network resources

*and*relay positions, across time. This formulation led to an efficient scheduling procedure for joint communication and relay mobility control, and enabled the development of robust and easily implementable,

*distributed*relay motion policies. An operationally important feature of this approach is that, although the system is optimized

*myopically*, the

*average*network QoS is

*nondecreasing*through time (in practice,

*increasing*), as long as the temporal dependence of the channel is sufficiently strong. In practice,

*our experiments have shown an*, compared to purely randomized relay motion (or no motion at all). This shows that strategic relay mobility can result in substantial performance gains, as far as enhancement of QoS is concerned.

**improvement of about 80% on the average network QoS at steady state**An important theoretical contribution, which resulted as a technical byproduct of the research outlined above, is the establishment of the so-called Interchangeability Principle (IP), for integral variational optimization problems, involving not necessarily semicontinuous random functions, defined on a complete probability space of arbitrary topological structure. This result may be seen as a useful, nontrivial extension of the well-known Theorem 14.60 of Rockafellar & Wets in the reference book “Variational Analysis”. Of course, the IP constitutes a fundamental tool in the development of many core results in decision analysis, such as the Bellman Equation, or the Minimum Mean Squared Error (MMSE) estimator, and finds interesting extensions to modern areas as well, such as sequential risk-averse optimization.

**Support:**

- NSF Grant CNS-1239188 (pi: Dr. Athina Petropulu)
- NSF Grant CCF-1526908 (as Senior Personnel, pi: Dr. Athina Petropulu, co-pi: Dr. Wade Trappe)

**Selected Publications:**

- D. S. Kalogerias and A. P. Petropulu, “Spatially Controlled Relay Beamforming,”
*IEEE Transactions on Signal Processing*, to appear, 2018. - D. S. Kalogerias and A. P. Petropulu, “Spatially Controlled Relay Beamforming: 2-Stage Optimal Policies,” Working Extended Arxiv Preprint, May 2017.
- D. S. Kalogerias and A. P. Petropulu, “Enhancing QoS in Spatially Controlled Beamforming Networks via Distributed Stochastic Programming,”
*42nd IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2017)*, New Orleans, LA, USA, March 2017.

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# Approximate

Nonlinear Stochastic Filtering

Nonlinear stochastic filtering refers to problems in which a stochastic process, the so-called state, is partially observed by measuring another stochastic process, the observations, and the objective is to best estimate a function of the state, based on causal observations. Adopting the MMSE as the standard criterion, in most cases, the nonlinear filtering problem results in a dynamical system in the infinite dimensional space of measures, making the need for approximate solutions imperative.

I have studied the fundamental problem of globally approximating general MMSE optimal nonlinear filters, in discrete time. Under a potentially non-Markovian, conditionally Gaussian problem setting, I have shown convergence of appropriately defined approximate filters to the true optimal filter in a strong and well defined sense. In particular, convergence is compact in time and uniform in a completely characterized event of probability almost 1, providing a useful quantitative justification of Egorov’s Theorem, for the filtering problem at hand. The purpose of such result is to enable analysis of various approximate filtering techniques, under a common, canonical framework.

In particular, this research led to a novel convergence analysis of grid based, recursive, approximate filters of Markov processes observed in conditionally Gaussian noise. More specifically, for grid based filters based on the so called marginal state quantization, I introduced the notion of conditional regularity of stochastic kernels, which, to the best of my knowledge, constitutes the most easily verifiable and relaxed condition proposed, under which strong asymptotic optimality of the respective grid based filters is guaranteed, in the sense briefly described above. To the best of my knowledge, no such results exist for competing global techniques for approximate filtering (for instance, particle filters), thus showing a potential theoretical advantage of the grid based approach.

Along the lines of the above works, several related studies have been pursued. Those include, in particular, a detailed stability analysis of distributed nonlinear state estimation in Gaussian-Finite hidden Markov models, implemented via the Alernating-Direction-Method-of-Multipliers (ADMM), a well known parallel optimization procedure in mathematical optimization.

**Support:**

- NSF Grant CNS-1239188 (pi: Dr. Athina Petropulu)
- NSF Grant CCF-1526908 (as Senior Personnel, pi: Dr. Athina Petropulu, co-pi: Dr. Wade Trappe)

**Selected Publications:**

- D. S. Kalogerias and A. P. Petropulu, “Uniform ε-Stability of Distributed Nonlinear Filtering over DNAs: Gaussian-Finite HMMs,”
*IEEE Transactions on Signal & Information Processing over Networks, (Special Issue on Inference & Learning over Networks)*, vol. 2, no. 4, pp. 461 – 476, December 2016. - D. S. Kalogerias and A. P. Petropulu, “Grid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality,”
*IEEE Transactions on Signal Processing*,

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# Matrix Completion in

Colocated MIMO Radar

I have developed useful results on the application of low rank Matrix Completion (MC) in a Multiple Input Multiple Output (MIMO) Radar setting, employed as a means for effectively reducing the volume of data required for target detection and estimation. I studied the applicability of MC on the type of data matrices appearing in colocated MIMO radar. In particular, for the classical case of a uniform linear array, I showed for the first time that the coherence of the data matrix is asymptotically optimal with respect to the number of antennas. As a result, the data matrix is provably recoverable via MC using a subset of its entries of minimal cardinality. These results were subsequently generalized to the case of an arbitrary 2-dimensional array, providing more general but yet easy to use sufficient conditions, ensuring low matrix coherence.

**Support:**

- NSF Grant CNS-1239188 (pi: Dr. Athina Petropulu)
- ONR Grant N00014-12-1-0036 (pi: Dr. Athina Petropulu)

**Selected Publications:**

- D. S. Kalogerias and A. P. Petropulu, “Matrix Completion in Colocated MIMO Radar: Recoverability, Bounds & Theoretical Guarantees,”
*IEEE Transactions on Signal Processing*, vol. 62, no. 2, pp. 309 – 321, January 2014. - D. S. Kalogerias and A. P. Petropulu, “MC-MIMO Radar: Recoverability and Performance Bounds,”
*1st IEEE Global Conference on Signal and Information Processing (GlobalSIP 2013)*, Austin, TX, USA, December 2013.