POLYNOMIAL TIME HEURISTIC OPTIMIZATION METHODS APPLIED TO PROBLEMS IN COMPUTATIONAL FINANCE

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POLYNOMIAL TIME HEURISTIC OPTIMIZATION METHODS
APPLIED TO PROBLEMS IN COMPUTATIONAL
FINANCE Ph.D. dissertation of Fogarasi Norbert,
M.Sc. Supervisor Dr. Levendovszky János, D.
Sc. Doctor of the Hungarian Academy of Sciences
Department of Telecommunications Budapest
University of Technology and Economics Budapest,
2014 February 25
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Outline of Presentation

• Introduction
• Computational Finance and Recent Industry Trends
• My contribution
• Thesis Group I. Mean reverting portfolio
  selection
• Thesis Group II. Optimal scheduling on identical
  machines
• Summary of results and contributions
• Questions and Answers

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Computational Finance and Recent Industry Trends

• Relatively new branch of computer science
  (Markowitz 1950s Modern Portfolio Theory. Nobel
  Prize in 1990)
• Numerical methods and algorithms with huge focus
  on applicability (quantitative study of markets,
  arbitrage, options pricing, mortgage
  securitization)
• Recent focus Algorithmic trading, quantitative
  investing, high frequency trading
• Post the 2008 financial crisis
• Regulatory pressure (timely reporting,
  transparency)
• High-frequency trading (flash crashes)
• Unprecedented focus on cost and efficiency
• Finding quick (polynomial time) approximate
  solutions to difficult (exponential, NP hard)
  problems is a key focus

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Computational Finance Open Issues
Challenges

• My Contribution

Polynomial time approximation using stochastic
optimization
Real-time portfolio identification
Overnight Monte-Carlo risk calculation scheduling
Polynomial time heuristic scheduling algorithms
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My Contribution

• Finding polynomial time approx solutions to NP
  hard problems
• Mean Reverting Portfolio selection (Thesis Group
  I)
• Task Scheduling on Identical machines (Thesis
  Group II)
• Show measurable improvement to existing
  approximate methods
• Prove practical applicability in real world
  settings
• Have very quick runtime characteristics for high
  frequency trading, timely regulatory reporting
  and hardware cost savings
• 5 refereed journal publications, 1 conference
  presentation
• 1. Fogarasi, N., Levendovszky, J. (2012) A
  simplified approach to parameter estimation and
  selection of sparse, mean reverting portfolios.
  Periodica Polytechnica, 56/1, 21-28.
• 2. Fogarasi, N., Levendovszky, J. (2012) Improved
  parameter estimation and simple trading algorithm
  for sparse, mean-reverting portfolios. Annales
  Univ. Sci. Budapest., Sect. Comp., 37, 121-144.
• 3. Fogarasi, N., Tornai, K., Levendovszky, J.
  (2012) A novel Hopfield neural network approach
  for minimizing total weighted tardiness of jobs
  scheduled on identical machines. Acta Univ.
  Sapientiae, Informatica, 4/1, 48-66.
• 4. Tornai, K., Fogarasi, N., Levendovszky, J.
  (2013) Improvements to the Hopfield neural
  network solution to the total weighted tardiness
  scheduling problem. Periodica Polytechnica, 57/1,
  1-8.
• 5. Fogarasi, N., Levendovszky, J. (2013) Sparse,
  mean reverting portfolio selection using
  simulated annealing. Algorithmic Finance, 2/3-4,
  197-211.
• 6. Fogarasi, N., Levendovszky, J. (2012)
  Combinatorial methods for solving the generalized
  eigenvalue problem with cardinality constraint
  for mean reverting trading. 9th Joint Conf. on
  Math and Comp. Sci. February 2012 Siofok, Hungary

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Summary of numerical results on real world
problems
Field Real world problem Average performance of traditional approaches Average performance of the proposed new method Impact on computational finance (improvement in percentage)
Portfolio optimization Convergence trading on US SP 500 stock data 11.6 (SP 500 index return) 34 22.4
Schedule optimization Morgan Stanley overnight scheduling problem 24709 (LWPF performance) 22257 (PSHNN performance) 10
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Thesis Group I. Mean reverting portfolio
selection

• Modern Portfolio Theory (MPT) maximize expected
  return for a given amount of risk
• Profitability vs. Predictability

• Mean-reverting portfolios have a large degree of
  predictability
• Therefore, we can develop profitable convergence
  trading strategies (35 annual return on
  portfolio selected from SP500)

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Intuitive task description
My contribution Developing novel algorithms for
identifying mean reverting portfolios with
cardinality constraints, trading and performance
analysis
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Thesis Group I. Problem Description
How to identify mean reverting portfolios based
on multivariate historical time series?
Constraint Sparse portfolio (limited transaction
costs, easier to understand/interpret
strategy) dAspremont, A.(2011) Identifying small
mean-reverting portfolios. Quantitative Finance,
113, 351-364 (Ecole Polytechnique, Paribas
London, Phd-Stanford, Postdoc-Berkeley, Princeton)
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Thesis Group I. Summary
Thesis I.1
New numerical method for estimating covariance matrix of VAR1 process
Periodica Polytechnica 2011
Thesis I.2
Adopted simulated annealing to probl of maximizing mean reversion under cardinality constraint
Algorithmic Finance 2013
Thesis I.3
Novel mean estimation technique for O-U processes using pattern matching
Annales Univ Sci Bp 2012
Thesis I.4
Simple trading strategy based on decision theoretic formulation
Joint Conf on Math and Comp Sci 2012
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Thesis Group I. The model
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The discrete model - VAR(1)First degree vector
autoregressive process
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Optimal portfolio as a generalized eigenvalue
problem
 
 
Problem develop a fast solution to the
generalized eigenvalue problem under the
cardinality constraint NP hard
?Poly time ??
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Thesis I.1 Estimation of Model Parameters

• Given nxT historical VAR(1) data st we need to
  estimate A, K (covar matrix of W) and G (covar
  matrix of st)
• A and K can be estimated using max likelihood
• G can be estimated using sample covariance.
  Classical research focuses on regularization
  techniques (Dempster 1972, Banerjee et al 2008,
  dAspremont et al 2008, Rothman et al 2008)
• My novel approach use sample covariance and an
  iterative recursive estimate in tandem to
  approximate G.

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Thesis I.1 Estimation of covariance

• From definition of VAR(1), we have the Lyapunov
  relationship in the stationary case
• However, the solution may be non-positive
  definite so we introduce a numerical method that
  ensures positive definiteness
• Start with G(0)sample covariance
• Also gives a goodness of model fit
• 0 for generated VAR(1) data, shows how well
  VAR(1) assumption works for real data.

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Thesis I.1 Numerical results
vs. t for n8, s0.1, 0.3, 0.5,
generating 100 independent time series for each t
and plotting the average norm of error
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Cardinality reduction by exhaustive search
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Polynomial Time Heuristic Approaches

• Greedy Method (dAspremont 2011) On each
  iteration, consider adding all remaining n-k
  dimensions and choose the one that yields the
  largest max eigenvalue.
• Truncation Method (Fogarasi et al 2012) Compute
  unconstrained solution then use k heaviest
  dimensions to solve the constrained problem.
  Super fast heuristic (only 2 eigenvalue
  computations)

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Greedy Solution (dAspremont 2011)

• Simple iterative heuristic that works very well
  in practice. DAspremont couldnt consistently
  outperform it.
• Let Ik be the set of indices belonging to the k
  non-zero components of x.
• On each iteration, we consider adding all
  remaining n-k dimensions and we choose the one
  that yields the largest max eigenvalue.
• Amounts to solving (n - k) generalized eigenvalue
  problems of size k 1. Polynomial runtime

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Thesis I.2 Application of SA by random
projection

• Restrict the portfolio vector x to have only
  integer values
• Consider the Energy function to be minimized
• At each step of the algorithm, we consider a
  neighboring state w' of the current state w and
  decide between moving or staying
• As T is decreased (cooling), above has been
  proven to converge (in distribution) to optimal
  solution.

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Thesis I.2 Application of SA by random
projection

• Cardinality constraint can easily be built into
  the neighbor function
• Starting point can be selected as Greedy solution
  
• Memory feature can be built in to ensure solution
  is at least as good as starting point
• Periodic revert to starting point improves
  performance
• Cooling schedule can be set to be fast enough for
  the specific application
• Procedure can be stopped at any point or an
  adaptive stopping condition has been developed.

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Thesis I.2 Numerical Results

• For n10, k5 Greedy and SA find theoretical best
  in 70 of cases, but in 11 of the remaining 30,
  SA outperforms Greedy.
• For larger problem sizes, SA performs even better
  (eg. for n20, k10 it outperforms Greedy 25 of
  the cases)

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Thesis I.2 Runtime Analysis

• Truncation method sub-second portfolio
  selection, can be used in real-time algorithmic
  trading
• Greedy seconds to compute, can be used in
  intraday trading
• Simulated Annealing minutes to compute, improves
  upon Greedy, can be used to finetune intraday
  trading
• Exhaustive impractical for ngt20, can be used for
  low frequency trading

CPU runtime (in seconds) versus total number of
assets n, to compute a full set of sparse
portfolios, with cardinality ranging from 1 to n
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Thesis I.3 Portfolio mean estimation

• Given historical portfolio valuations pt and
  assuming it follows O-U process, estimate µ.
• Classical methods in literature
• Sample mean estimate
• Least squares regression
• Max likelihood estimator (numerically complex)
• I developed a novel mean estimation method based
  on pattern matching and decision theory

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Thesis I.3 Novel portfolio parameter estimation
using pattern matching

• Starting from definition of Ornstein-Uhlenbeck
  process
• Taking expected value of the above
• Typical tendencies of µ(t) below (idea portf
  value without noise given µ(0) and long term µ)

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Thesis I.3 Portfolio mean estimation

• Use pt and max likelihood estimation techniques
  to decide which pattern they match the most, and
  determine long term µ
• where U is the time correlation matrix of pt
• This estimate is more accurate than sample mean
  and more resilient to small ? than linear
  regression.

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Thesis I.3 Portfolio mean estimation
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Thesis I.4 Simple Convergence Trading Model

• We are deciding whether µ(t)lt µ by only observing
  p(t) using an approach based on decision theory
• We can use this simplified model to prove the
  economic viability of our algorithms and compare
  them to each other.

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Thesis I.4 Simple Convergence Trading Model
As a result, for a given rate of acceptable error
e , we can select an a for which
Thus the trading strategy can then be summarized
as follows
Observed sample Accepted hypothesis Error probability Action (Cash / Portfolio)
Buy / Hold
No Action / Sell
No Action / Sell
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Convergence Trading Simulation on synthetic data

• Generated VAR(1) sequence
• All four methods produced average profits of the
  same order of magnitude and the distribution of
  trading gains was very similar.
• The exhaustive method produced mean reversion
  coefficients on average 15 times those produced
  by the truncation method and 3 times those
  produced by the greedy method and simulated
  annealing
• Profits reached by this simple convergence
  trading strategy are not directly proportional to
  the lambda produced by the portfolio selection
  method.
• Implies that more complex objective functions
  could be developed

Total number of assets (n) 10
Cardinality constraint (k) 5
Number of data points for portfolio selection (T) 20
Number of data points in trading window 250 ( of trading days in a year)
Number of simulations ran (rep) 2000
Initial cash 100
Maximum cash 14M
Percentage of repeats profitable 97
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Convergence Trading Simulation

• Typical example of convergence trading over 250
  time steps on a sparse mean-reverting portfolio..
  A profit of 1440 was achieved with an initial
  cash of 100 having made 85 transactions.

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Convergence Trading Simulation Histogram of
profits generated by SA

• Histogram of profits achieved over 2000 different
  generated VAR(1) series by simulated annealing.

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Thesis Group I. SP500 Test

• Consider the 500 stocks that make up the SP500
  during 2009-2010 and select the K4 stock
  portfolio to maximize mean-reversion.
• Repeat for 250 trading days (1 year)
• SP500 went up by 11.6, our method generates 34
  return
• Minimum, maximum, average and final portfolio
  values starting from 100.

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Thesis Group I. Conclusions and directions for
future research

• Novel parameter estimation techniques introduced
• Successfully applied simulated annealing with
  random projections to meet cardinality
  constraints
• Outperformed greedy method in 10 of the cases
  for 10 assets, 25 for 20 assets on generated
  data (good scaling features)
• Simple trading model developed for testing
  economic viability, proved significant
  improvement over other methods
• More factors need to be considered for objective
  function more closely linked to trading profits
• Simulated Annealing approach can be used for
  these too!
• More complex trading model can be developed to
  increase profits

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Thesis Group II. Optimal Scheduling

• Scheduling problems manufacturing,
  pharmaceutical, biological, financial
  computations.
• Complex portfolios are evaluated and risk managed
  using Monte-Carlo simulations at many financial
  institutions (eg. Morgan Stanley)
• Each night a changed portfolio needs to be
  evaluated/risk managed with new market data/model
  parameters
• Need a quick way to schedule 10000s of jobs on
  10000s of machines in a near optimal way
• Why? 10M/year spend on hardware, timely response
  to clients and regulators regarding portfolio
  values and VaR.
• My novel method saved 53 minutes on top priority
  jobs running for 12 hours overnight, compared to
  the next best heuristic.

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Thesis Group II. Problem Formulation

• Scheduling jobs on a finite number (V) of
  identical processors under constraints on the
  completion times
• Given n users/jobs of sizes
• Cutoff times
• Weights/priorities
• Scheduling matrix
• Where if job i is processed at
  time step j.
• Jobs can stop/restart on different machine
  (preemption)
• For example V2, n3, x2,3,1, K3,3,3.

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Thesis Group II. Problem Formulation

• Define Tardiness as
• where is the finishing time of job i as
  per C.
• Minimizing Total Weighted Tardiness (TWT) is
  stated as
• Under the following constraints
• For example V2, n3, x2,3,2, K3,3,3,
  w3,2,1 All jobs cannot complete before their
  cutoff times, but the optimal TWT solution is

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Heuristic Approaches to TWT

• 1990 Du and Leung prove that TWT is NP-hard
• 1979 Dogramaci, Sulkis simple heuristic
• 1983 Rachamadugu myopic heuristic, compares to
  Earliest Due Date (EDD) and WSPT (Weighted
  Shortest Processing Time)
• 1998 Azizoglu branch and bound heuristic, too
  slow gt 15 jobs
• 1994 Koulamas KPM algorithm
• 2000 Armentano tabu search
• 1995 Guinet simulated annealing, lower bound
• 2002 Sen, 2008 Biskup Surveys of existing
  methods
• 2000 Artificial Neural Network approach to
  scheduling problems
• 2004 Maheswaran Hopfield Neural Network
  approach to single machine TWT on a specific
  10-job problem.

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Thesis Group II. Optimal Scheduling
Thesis II.1 Thesis II.2 Thesis II.3 Thesis II.4
I converted TWT problem to quadratic form including the constraints with heuristic constants I applied the Hopfield Neural Net (HNN) and found approximate solutions in polynomial time I showed that HNN solution outperforms other simple heuristics on large set of random problems I improved HNN by intelligent selection of starting point and random perturbations
Acta Univ Sapientiae 2012 Acta Univ Sapientiae 2012 Acta Univ Sapientiae 2012 Periodica Polytechnica 2013
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Thesis II.1 TWT to QP

• HNN are a recursive Neural Network which are good
  for solving quadratic optimization problems in
  the form
• Has been applied successfully for finding good
  approximate solutions to the Travelling Salesman
  problem.
• Our task is to transform the TWT to a quadratic
  optimization problem.

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Thesis II.1 TWT to QP

• Move constraints to objective function
• Each member of the above addition can be
  converted to quadratic Lyapunov form separately
  to bring the expression into the form

R
R
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Thesis II.1 TWT to QP

• Results of the matrix conversions

R
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Thesis II.2 Applying HNN

• Hopfield (1982) proved that the recursion
• converges to its fixed point, so minimizes a
  quadratic Lyapunov function
• I implemented this in MATLAB, including
  systematic selection of the heuristic constants
  a,ß and ?. I also developed algorithms to
  validate and correct the resulting schedule
  matrix if needed.

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Thesis II.3 HNN outperforms other simple
heuristics

• For each problem size ( of jobs) 100 random
  problems were generated and the average TWT was
  computed and plotted

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Thesis II.3 HNN outperforms other simple
heuristics

• Outperformance is consistent over a broad
  spectrum of problems over simple heuristics in
  literature (LWPF Largest Weighted Process
  First, WSPT Weighted Shortest Processing Time,
  EDD Earliest Due Date)

Job size 5 10 15 20 30 40 50 75 100
outperf 99.9 100 100 99.5 99.2 99.6 99.3 98.6 98.8
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Thesis II.4 Further improving HNN

• Smart HNN (SHNN)
• Use the resultof Largest Weighted Path First
  (LWPF) as starting point for HNN rather than
  random starting points
• Speeds up HNN due to single starting point, but
  still require multiple iterations due to setting
  of heuristic constants.
• Perturbed Smart HNN (PSHNN)
• Consider random perturbations of LWPF as starting
  point to HNN, in order to avoid getting stuck in
  local minima

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Thesis II.4 Further improving HNN

• Perturbed Largest Weighted Path First (PLWPF)
• Simple, but surprisingly well performing heuristi
• The idea is to avoid getting stuck in local
  minima by trying starting points near LWPF
  solution

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Thesis II.4 Further improving HNN

• For small job sizes, we compare performance to
  the theoretical best exhaustive search over 100
  randomly generated problems per job size

• PSHNN consistently outperforms other methods, but
  there is room for improvement

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Thesis II.4 Further improving HNN

• For small job sizes, we compare performance to
  the theoretical best exhaustive search over 100
  randomly generated problems per job size

• PSHNN outperforms other methods by increasing
  margin as job size grows

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Thesis Group II. Practical Application

• Monte Carlo simulation based risk calculations
  scheduling at Morgan Stanley overnight for
  trading and regulatory reporting
• 100 portfolios, 556 jobs, 792 seconds average
  size
• 7 improvement over HNN, 10 over LWPF (best
  method in literature prior to my study).
• 53 minutes saved on top 3 priority jobs compated
  to next best heuristic

Weight 3 4 5 6 7 8 9 10 SUM Increment to PSHNN
PSHNN 4401 11116 4020 1620 1092 8 0 0 22257 0
PLWPF 3513 9624 5130 1788 490 312 2304 190 24019 5
HNN 4404 11040 4735 1824 1092 456 468 0 24019 7
LWPF 4404 11140 5470 2472 1183 40 0 0 24709 10
EDD 4401 9940 1770 636 1134 464 22752 1430 42527 48
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Summary of numerical results on real world
problems
Field Real world problem Average performance of traditional approaches Average performance of the proposed new method Impact on computational finance (improvement in percentage)
Portfolio optimization Convergence trading on US SP 500 stock data 11.6 (SP 500 index return) 34 22.4
Schedule optimization Morgan Stanley overnight scheduling problem 24709 (LWPF performance) 22257 (PSHNN performance) 10
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Summary of my Contribution
Managed to find a generic approach to
approximating NP hard problems in polynomial time
using heuristic methods
Proved the practical effectiveness and
applicability on real world problems for 2 very
difficult open problems
This can speed up financial calculations and
their scheduling

• Provides faster, more timely data to banks,
  clients and financial regulators ? improves
  society as a whole

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Thank You For Your Attention!
Questions and Answers
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Questions and Answers

• In the VAR(1) parameter estimation, could you use
  a stability theorem in the solution of the
  Lyapunov equation to ensure G is positive
  definite, rather than using the numerical method
  suggested?
• The Lyapunov equation is
• If K and A are positive definite and A has
  eigenvalues with modulus lt 1 then G is guaranteed
  to be positive definite
• When estimating A from real world data using
  maximum likelihood, I found that the best
  estimate often had eigenvalues with modulus gt 1,
  and G became non-positive-definite, hence my
  suggested method.
• One could research new ways to estimate A to
  ensure eigenvalue moduli are lt 1.