# Panos Parpas's research while affiliated with Imperial College London and other places

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## Publications (73)

A deep learning approach for the approximation of the Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) associated to the Nonlinear Quadratic Regulator (NLQR) problem. A state-dependent Riccati equation control law is first used to generate a gradient-augmented synthetic dataset for supervised learning. The resulting model becomes a w...

We consider the problem of minimizing a convex function that depends on an uncertain parameter $\theta$. The uncertainty in the objective function means that the optimum, $x^*(\theta)$, is also a function of $\theta$. We propose an efficient method to compute $x^*(\theta)$ and its statistics. We use a chaos expansion of $x^*(\theta)$ along a trunca...

Recovering a low-rank matrix from highly corrupted measurements arises in compressed sensing of structured high-dimensional signals (e.g., videos and hyperspectral images among others). Robust principal component analysis (RPCA), solved via principal component pursuit (PCP), recovers a low-rank matrix from sparse corruptions that are of unknown val...

The mirror descent algorithm is known to be effective in applications where it is beneficial to adapt the mirror map to the underlying geometry of the optimization model. However, the effect of mirror maps on the geometry of distributed optimization problems has not been previously addressed. In this paper we propose and study exact distributed mir...

We consider the problem of minimising a strongly convex function that varies with an uncertain parameter $\theta$. This uncertain parameter means the optimum is also a function of $\theta$, and we aim to learn about this function and its statistics. We use chaos expansions, a technique that has been used for stochastic approximation, and use gradie...

The analysis of second-order optimization methods based either on sampling, randomization or sketching has two serious shortcomings compared to the conventional Newton method. The first shortcoming is that the analysis of the iterates is not scale-invariant, and even if it is, restrictive assumptions are required on the problem structure. The secon...

In this paper, we consider the problem of parameter estimation for a stochastic McKean-Vlasov equation, and the associated system of weakly interacting particles. We first establish consistency and asymptotic normality of the offline maximum likelihood estimator for the interacting particle system in the limit as the number of particles $N\rightarr...

An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent repl...

An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. However, to converge the step-size must decrease exponentially slow, and therefore this approach is not useful in pra...

Unreported partially/fully closed valves or other types of pipe blockages in water distribution networks result in unexpected energy losses within the systems, which we also refer to as faults. We investigate the problem of detection and localization of such faults. We propose a novel optimization-based method, which relies on the solution of a non...

Inspired by multigrid methods for linear systems of equations, multilevel optimization methods have been proposed to solve structured optimization problems. Multilevel methods make more assumptions regarding the structure of the optimization model, and as a result, they outperform single-level methods, especially for large-scale models. The impress...

Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection between PDEs and systems of Forward-Backward Stochastic Differential Equations (FBSDEs) enables the use of advance...

Empirical risk minimization is recognized as a special form in standard convex optimization. When using a first order method, the Lipschitz constant of the empirical risk plays a crucial role in the convergence analysis and stepsize strategies for these problems. We derive the probabilistic bounds for such Lipschitz constants using random matrix th...

We consider the problem of minimising functions represented as a difference of lattice submodular functions. We propose analogues to the SupSub, SubSup and ModMod routines for lattice submodular functions. We show that our majorisation-minimisation algorithms produce iterates that monotonically decrease, and that we converge to a local minimum. We...

An open problem in machine learning is whether flat minima generalize better and how to compute such minima efficiently. This is a very challenging problem. As a first step towards understanding this question we formalize it as an optimization problem with weakly interacting agents. We review appropriate background material from the theory of stoch...

The links between optimal control of dynamical systems and neural networks have proved beneficial both from a theoretical and from a practical point of view. Several researchers have exploited these links to investigate the stability of different neural network architectures and develop memory efficient training algorithms. We also adopt the dynami...

This paper analyzes the relation between different orders of the Lasserre hierarchy for polynomial optimization (POP). Although for some cases solving the semidefinite programming relaxation corresponding to the first order of the hierarchy is enough to solve the underlying POP, other problems require sequentially solving the second or higher order...

Recovering a low-rank matrix from highly corrupted measurements arises in compressed sensing of structured high-dimensional signals (e.g., videos and hyperspectral images among others). Robust principal component analysis, solved via principal component pursuit (PCP), recovers a low-rank matrix from sparse corruptions that are of unknown value and...

We propose a multigrid approach for the global optimization of polynomial optimization problems with sparse support. The problems we consider arise from the discretization of infinite dimensional optimization problems, such as PDE optimization problems, boundary value problems, and some global optimization applications. In many of these application...

Supporting Information.

Supporting Information.

Supporting Information.

We calculate arbitrarily tight upper and lower bounds on an unconstrained control, linear-quadratic, singularly perturbed optimal control problem whose exact solution is computationally intractable. It is well known that for the aforementioned problem, an approximate solution $\bar{V}^N(\epsilon)$ can be constructed such that it is asymptotically e...

We present a methodology for bounding the error term of an asymptotic solution to a singularly perturbed optimal control (SPOC) problem whose exact solution is known to be computationally intractable. In previous works, reduced or computationally tractable problems that are no longer dependent on the singular perturbation parameter $\epsilon$, wher...

Response to antidepressant (AD) treatment may be a more polygenic trait than previously hypothesized, with many genetic variants interacting in yet unclear ways. In this study we used methods that can automatically learn to detect patterns of statistical regularity from a sparsely distributed signal across hippocampal transcriptome measurements in...

Large-scale nonsmooth convex optimization is a common problem for a range of computational areas including machine learning and computer vision. Problems in these areas contain special domain structures and characteristics. Special treatment of such problem domains, exploiting their structures, can significantly reduce the computational burden. In...

Significant changes in the power generation mix are posing new challenges for the balancing systems of the grid. Many of these challenges are in the secondary electricity grid regulation services and could be met through demand response (DR) services. We explore the opportunities for a water distribution system (WDS) to provide balancing services w...

The operation of water distribution networks (WDN) with a dynamic topology is a recently pioneered approach for the advanced management of District Metered Areas (DMAs) that integrates novel developments in hydraulic modeling, monitoring, optimization, and control. A common practice for leakage management is the sectorization of WDNs into small zon...

Composite convex optimization models arise in several applications, and are
especially prevalent in inverse problems with a sparsity inducing norm and in
general convex optimization with simple constraints. The most widely used
algorithms for convex composite models are accelerated first order methods,
however they can take a large number of iterat...

The operation of pump systems in water distribution systems (WDS) is commonly the most expensive task for utilities with up to 70% of the operating cost of a pump system attributed to electricity consumption. Optimisation of pump scheduling could save 10-20% by improving efficiency or shifting consumption to periods with low tariffs. Due to the com...

Stochastic programming models are large-scale optimization problems that are used to facilitate decision making under uncertainty. Optimization algorithms for such problems need to evaluate the expected future costs of current decisions, often referred to as the recourse function. In practice, this calculation is computationally difficult as it req...

This paper presents a novel concept of adaptive water distribution networks with dynamically reconfigurable topology for optimal pressure control, leakage management and improved system resilience. The implementation of District Meter Areas (DMAs) has greatly assisted water utilities in reducing leakage. DMAs segregate water networks into small are...

Long-term planning for electric power systems, or capacity expansion, has traditionally been mod-eled using simplified models or heuristics to approximate the short-term dynamics. However, cur-rent trends such as increasing penetration of intermittent renewable generation and increased de-mand response requires a coupling of both the long and short...

Singular perturbation techniques allow the derivation of an aggregate model whose solution is asymptotically optimal for Markov decision processes with strong and weak interactions. We develop an algorithm that takes advantage of the asymptotic optimality of the aggregate model in order to compute the solution of the original model. We derive condi...

We investigate the feasibility of detecting host-level CPU contention from inside a guest virtual machine (VM). Our methodology involves running benchmarks with deterministic and randomized execution times inside a guest VM in a private cloud testbed. Simultaneously, using the recently proposed COCOMA tool, we expose the guest VM to host-level CPU...

Existing complexity results in stochastic linear programming using the Turing model depend only on problem dimensionality. We apply techniques from the information-based complexity literature to show that the smoothness of the recourse function is just as important. We derive approximation error bounds for the recourse function of two-stage stochas...

We present a numerical method for finite-horizon stochastic optimal control models. We derive a stochastic minimum principle (SMP) and then develop a numerical method based on the direct solution of the SMP. The method combines Monte Carlo pathwise simulation and non-parametric interpolation methods. We present results from a standard linear quadra...

To evaluate system capacity, past works on MultipleInput-Multiple-Output (MIMO) systems with mutually interfering links have focused on maximizing the sum of mutual information as the objective criterion. Since the ultimate goal of a MIMO system is to support network applications used by consumers, we consider the non-concave sigmoid utility functi...

Analyses of global climate policy as a sequential decision under uncertainty have been severely restricted by dimensionality and computational burdens. Therefore, they have limited the number of decision stages, discrete actions, or number and type of uncertainties considered. In particular, other formulations have difficulty modeling endogenous or...

Markov Random Fields (MRF) minimization is a well-known problem in computer vision. We consider the augmented dual of the MRF minimization problem and develop a Mirror Descent algorithm based on weighted Entropy and Euclidean Projection. The augmented dual problem consists of maximizing a non-differentiable objective function subject to simplex and...

Decisions on whether to invest in new power system infrastructure can have farreaching consequences. The timely expansion of generation and transmission capacities is crucial for the reliability of a power system and its ability to provide uninterrupted service under changing market conditions.We consider a local (e.g., regional or national) power...

We consider the problem of finding the minimum of a real-valued multivariate polynomial function constrained in a compact
set defined by polynomial inequalities and equalities. This problem, called polynomial optimization problem (POP), is generally
nonconvex and has been of growing interest to many researchers in recent years. Our goal is to tackl...

We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm
follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility
of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analy...

Many decision models can be formulated as continuous minimax problems. The minimax framework injects robustness into the model. It is a tool that one can use to perform worst-case analysis, and it can provide considerable insight into the decision process. It is frequently used alongside other methods such as expected value optimization in order to...

The classical Markowitz approach to portfolio selection is compromised by two major shortcomings. First, there is considerable model risk with respect to the distribution of asset returns. Particularly, mean returns are notoriously difficult to estimate. Moreover, the Markowitz approach is static in that it does not account for the possibility of p...

We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in [1, 2] that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decomp...

We propose a stochastic algorithm for the global optimization of chance constrained problems. We assume that the probability measure with which the constraints are evaluated is known only through its moments. The algorithm proceeds in two phases. In the first phase the probability distribution is (coarsely) discretized and solved to global optimali...

We discuss the global optimization of the higher order moments of a portfolio of financial assets. The proposed model is an
extension of the celebrated mean variance model of Markowitz. Asset returns typically exhibit excess kurtosis and are often
skewed. Moreover investors would prefer positive skewness and try to reduce kurtosis of their portfoli...

We consider the problem of finding the minimum of a real-valued multivariate polynomial function constrained in a compact set defined by polynomial inequalities and equalities. This problem, called polynomial optimization problem (POP), is generally nonconvex and has been of growing interest to many researchers in recent years. Our goal is to tackl...

Recently, given the first few moments, tight upper and lower bounds of the no arbitrage prices can be obtained by solving semidefinite programming (SDP) or linear programming (LP) problems. In this paper, we compare SDP and LP formulations of the European-style options pricing problem and prefer SDP formulations due to the simplicity of moments con...

The Markowitz Mean Variance model (MMV) and its variants are widely used for portfolio selection. The mean and covariance
matrix used in the model originate from probability distributions that need to be determined empirically. It is well known
that these parameters are notoriously difficult to estimate. In addition, the model is very sensitive to...

We generalize a smoothing algorithm for finite min–max to finite min–max–min problems. We apply a smoothing technique twice,
once to eliminate the inner min operator and once to eliminate the max operator. In mini–max problems, where only the max
operator is eliminated, the approximation function is decreasing with respect to the smoothing paramete...

We consider the problem of efficient resource allocation in a grid comput- ing environment. Grid computing is an emerging paradigm that allows the sharing of a large number of a heterogeneous set of resources. We propose an auction mechanism for decentralized resource allocation. The problem is modeled as a multistage stochastic programming problem...

A discretization scheme for a portfolio selection problem is discussed. The model is a benchmark relative, mean-variance optimization
problem in continuous time. In order to make the model computationally tractable, it is discretized in time and space. This
approximation scheme is designed in such a way that the optimal values of the approximate pr...

Background
Problem Statement
Methods
Nested Benders Decomposition (NBD)
Augmented Lagrangian Decomposition (ALD)
Numerical Experiments
References

Abstract Duality gaps in optimization problems arise because of the nonconvex- ities involved in the objective function or constraints. The Lagrangian dual of a nonconvex optimization problem can also be viewed as a two person zero sum game. From this viewpoint, the occurrence of du- ality gaps originate from the order in which the two players sele...

We study bounding approximations for a multistage stochastic program with expected value constraints. Two simpler approximate stochastic programs, which provide upper and lower bounds on the original problem, are obtained by replacing the original stochastic data process by finitely supported approximate processes. We model the original and approxi...

Background
Models
Scenario Generation
Portfolio Selection
Methods
A Stochastic Optimization Algorithm
Other Methods
References

Article Outline
Abstract
Background
Heuristic Foundations of the Method
Applications
Stochastic Methods for Global Optimization
Phase Transitions in Combinatorial Optimization
Worst Case Optimization
References
Abstract
The Laplace method has found many applications in the theoretical and applied study of optimization problems. It has been used to...

We consider decomposition approaches for the solution of multistage stochastic programs that appear in financial applications. In particular, we discuss the performance of two algorithms that we test on the mean-variance portfolio optimization problem. The first algorithm is based on a regularized version of Benders decomposition, and we discuss it...

A stochastic algorithm is proposed for the global optimization of nonconvex functions subject to linear constraints. Our method follows the trajectory of an appropriately defined Stochastic Differential Equation (SDE). The feasible set is assumed to be comprised of linear equality constraints, and possibly box constraints. Feasibility of the trajec...

We consider the global optimization of two problems arising from financial applications. The first problem originates from
the portfolio selection problem when high-order moments are taken into account. The second issue we address is the problem
of scenario generation. Both problems are non-convex, large-scale, and highly relevant in financial engi...

In this paper we discuss a basic framework for a grid computing market. It has long been argued that pricing of computer resources can act as a scheduling protocol. We take this idea to its natural conclusion by discussing the basic properties of such a model. We introduce agents that own computer resources on the grid. We allow the agents to trade...

It is becoming apparent that convex financial planning models are at times a poor approximation of the real world. More realistic, and more relevant models need to dispense with normality assumptions, and concavity of the utility functions to be optimized. Moreover, the problems are large scale but structured, consequently specialized algo-rithms h...

The Laplace method has found many applications in the theoretical and applied study of optimization problems. It has been used to study: the asymptotic behavior of stochastic algorithms, 'phase transitions' in combinatorial optimization, and as a smoothing technique for non- differentiable minimax problems. This article describes the theoretical fo...

We present a numerical method for the optimal control of systems with Markovian dynamics in continuous time. We first derive a Stochastic Minimum Principle (SMP) under the assumption that the set of admissible control laws are smooth. We then develop a numerical method based on the direct solution of the SMP by combining Monte Carlo simulation and...

## Citations

... It is well known that this non-reversible diffusion converges faster to π β , and admits a lower asymptotic variance, than its reversible counterpart (Hwang et al., 2005;Duncan et al., 2016). One can also consider a system of diffusions, interacting via a matrix A, which also exhibit improved convergence properties (Borovykh et al., 2021). Given these results, it is natural to ask whether is it possible to determine an optimal perturbation, or an optimal interaction matrix. ...

... Considering that computing a lowrank matrix usually involves SVD, the computational advantage that (1.2) brings is even more significant. It is worth mentioning that our idea behind PRPCA shares similarities with the multilevel algorithms for Compressed PCP proposed by Hovhannisyan et al. (2019), where the interpolation matrix is used to build connections between the original "fine" model and a smaller "coarse" model. Their work is mainly from computational perspective, but the principle behind is the same: by applying SVD in models of lowerdimension, the computational burden can be significantly reduced. ...

... These algorithms rely on non-negativity certificates provided by sum-of-squares polynomials (Shor, 1987;Lasserre, 2001;Parrilo, 2003). Despite various enhancements that exploit special structures and sparsity in constructing the SDP relaxations (De Klerk, 2010;Nie and Demmel, 2009;Nie, 2019), or that take advantage of solutions already calculated to find initial approximations for the solutions of the higher-order relaxations (Campos et al., 2019), often only the first few elements in the hierarchy end up being computationally tractable for large-scale problems. Available codes implementing this approach include GloptiPoly (Henrion et al., 2009) and SOSTools (Papachristodoulou et al., 2013). ...

... In addition, solving the exact optimization problem requires storage of the entire data in the main memory, which is not feasible for large-scale calcium imaging data sets. Thus, many variants of RPCA have been developed to increase the speed [10,27,28] or to process large data [8,19,24], but it remains a challenge to simultaneously achieve high speed and scalability. ...

... In this paper, motivated by the recent advances in multilevel optimisation algorithms [11,13,15,22,5,27], we propose a simple, yet generic and very effective multilevel approach for significantly reducing computational costs for many problems that require solving nuclear norm based oracles, including RPCA models such as the PCP and CPCP. The core of our proposed methodology is to construct and solve lower dimensional (coarse) models for each optimisation oracle and then lift its solution to the original problem dimension. ...

... The object features together with the interests of each stakeholder form the objective. While the criteria by following availability form constraints [70]. Some researchers try to juxtapose optimization with data mining, to overcome the weaknesses of optimization that cannot produce both data and features from the information space directly [33,71]. ...

... Multi-level smooth unconstrained optimization is the setting of [Nas00, GST08,WG09], that includes trust-region and line search optimization. The first multi-level optimization method that covers the nonsmooth convex composite case was introduced in [Par17]. It is a multi-level version of the well-known proximal gradient method. ...

... As our methodology provides definitive bounds on the solution for all values of , it both increases the amount of information available when considering the implementation of an approximate solution and produces a criterion for determining how good of an approximation V N yields. While the authors' previous work in [18] established upper and lower bounds on the solution to a control constrained, linearquadratic SPOC problem of the form ...

... [23] Other inflammatory candidate genes such as IL2, IL10, IL18, tumor necrosis factor alpha (TNF-α) and interferon-gamma (IFN-g) have received less attention, although having been previously related with major depressive disorder. [24,25] Taking into account that response to medication may be a polygenic and complex trait than previously hypothesized and distinct molecular mechanisms seem to modulate inflammatory pathways as stated above [26], studies should address the role of different cytokine genetic variants, as response to medication may be as complex as the pathology. Besides, recent studies have also shown that alterations in the methylation status that regulate gene expression of these polymorphisms may contribute to disease risk and prognosis [27,28], but few studies have examined whether inflammatory genes are differentially methylated in depression (Ryan's study in late life depression [29]), or the association of methylation status of such polymorphisms with treatment outcomes. ...

... The Mirror Descent method which originated in [80,81] and was later analyzed in [18], is a substantial generalization of the subgradient methods allowing to adjust, to some extent, the method to the possibly non-Euclidean geometry of the problem in question. Mirror Descent method extends the standard projected subgradient methods by employing a nonlinear distance function with an optimal step-size in the nonlinear projection step [74], and by replacing the Euclidean proximal term in (1.20) by a Bregman divergence (see Subsec. 1.2.3, below) [61] x k+1 = arg min ...