Introductory talk more technicities in @inproceedings{schoenauer:inria-00625855, hal_id = {inria-00625855}, url = {http://hal.inria.fr/inria-00625855}, title = {{A Rigorous Runtime Analysis for Quasi-Random Restarts and Decreasing Stepsize}}, author = {Schoenauer, Marc and Teytaud, Fabien and Teytaud, Olivier}, abstract = {{Multi-Modal Optimization (MMO) is ubiquitous in engineer- ing, machine learning and artificial intelligence applications. Many algo- rithms have been proposed for multimodal optimization, and many of them are based on restart strategies. However, only few works address the issue of initialization in restarts. Furthermore, very few comparisons have been done, between different MMO algorithms, and against simple baseline methods. This paper proposes an analysis of restart strategies, and provides a restart strategy for any local search algorithm for which theoretical guarantees are derived. This restart strategy is to decrease some 'step-size', rather than to increase the population size, and it uses quasi-random initialization, that leads to a rigorous proof of improve- ment with respect to random restarts or restarts with constant initial step-size. Furthermore, when this strategy encapsulates a (1+1)-ES with 1/5th adaptation rule, the resulting algorithm outperforms state of the art MMO algorithms while being computationally faster.}}, language = {Anglais}, affiliation = {TAO - INRIA Saclay - Ile de France , Microsoft Research - Inria Joint Centre - MSR - INRIA , Laboratoire de Recherche en Informatique - LRI}, booktitle = {{Artificial Evolution}}, address = {Angers, France}, audience = {internationale }, year = {2011}, month = Oct, pdf = {http://hal.inria.fr/inria-00625855/PDF/qrrsEA.pdf}, }

1. Hard optimization
(multimodal, adversarial,
noisy)
http://www.lri.fr/~teytaud/hardopt.odp
http://www.lri.fr/~teytaud/hardopt.pdf
(or Quentin's web page)
Acknowledgments: koalas, dinosaurs, mathematicians.
Olivier Teytaud olivier.teytaud@inria.fr
Knowledge also from A. Auger, M. Schoenauer.

2. The next slide is the most important
of all.
Olivier Teytaud
Inria Tao, en visite dans la belle ville de Liège

3. In case of trouble,
Interrupt me.
Olivier Teytaud
Inria Tao, en visite dans la belle ville de Liège

4. In case of trouble,
Interrupt me.
Further discussion needed:
- R82A, Montefiore institute
- olivier.teytaud@inria.fr
- or after the lessons (the 25 th
, not the 18th)
Olivier Teytaud
Inria Tao, en visite dans la belle ville de Liège

5. Rather than complex algorithms,
we'll see some concepts and some simple
building blocks for defining algorithms. Robust optimization
Noisy optimization
Warm starts Nash equilibria
Coevolution Surrogate models
Niching Maximum uncertainty
Clearing Monte-Carlo estimate
Multimodal optimization Quasi-Monte-Carlo
Restarts (sequential / parallel) Van Der Corput
Koalas and Eucalyptus Halton
Dinosaurs (and other mass extinction) Scrambling
Fictitious Play EGO
EXP3

6. I. Multimodal optimization
II. Adversarial / robust cases
III. Noisy cases

7. e.g. Schwefel's function
(thanks to
Irafm)

8. What is multimodal optimization ?
- finding one global optimum ?
(not getting stuck in a local minimum)
- finding all local minima ?
- finding all global minima ?

9. Classical methodologies:
1. Restarts
2. ``Real'' diversity mechanisms

10. Restarts:
While (time left >0)
{
run your favorite algorithm
}
(requires halting criterion + randomness – why ?)

11. Parallel restarts
Concurrently, p times:
{
run your favorite algorithm
}
==> still needs randomization

12. Parallel restarts
Concurrently, p times:
{
run your favorite algorithm
}
==> needs randomization
(at least in the initialization)

13. Random restarts Quasi-random restarts

14. Let's open a parenthesis...
(

15. The Van Der Corput sequence:
quasi-random numbers in dimension 1.
How should I write the nth point ?
(p=integer, parameter of the algorithm)

16. A (simple) Scrambled
Van Der Corput sequence:

17. Halton sequence:
= generalization to dimension d
Choose p1,...,pd
All pi's must be different (why ?).
Typically, the first prime numbers (why ?).

18. Scrambled Halton sequence:
= generalization to dimension d with scrambling
Choose p1,...,pd
All pi's must be different (why ?).
Typically, the first prime numbers (why ?).

19. Let's close the parenthesis...
(thank you Asterix)
)

20. Consider an ES (an evolution strategy = an
evolutionary algorithm in continuous domains).
The restarts are parametrized by
an initial point x and an initial step-size .
Which hypothesis on x and  do we need, so that the
restarts of the ES will find all local optima ?

21. X*(f)
= set of optima
If I start close enough to an optimum, with
small enough step-size, I find it.

22. Restart (RS) algorithm:

24. What do you suggest ?

25. Accumulation
of a sequence

26. THEOREM:

27. THEOREM:
( (1) ==> (2); do you see why ?)

28. Ok, step-sizes should accumulate around 0,
and the xi's should be dense.
X(i) = uniform random = ok
and Or something decreasing;
But not a constant.
sigma(i) = exp( Gaussian )
are ok. (Why ?)
Should I do something more sophisticated ?

29. I want a small number of restarts.
A criterion: dispersion (some maths
should be developped around why...).

30. for random points
==> small difference
==> the key point is more the decreasing (or small) step-size

31. Classical methodologies:
1. Restarts
2. ``Real'' diversity mechanisms

32. 7
~ -7x10 years ago
Highly optimized.
Hundreds of millions of
years on earth.

33. 7
~ -7x10 years ago
Died!
Not adapted to
the new environment.

34. 7
~ -7x10 years ago
Survived thanks
to niching.

35. Eating Eucalyptus
is dangerous for
health (for many animals).
But not for koalas.
Also they sleep
most of the day.
They're original animals.
==> This is an example of Niching.
==> Niching is good for diversity.

36. USUAL SCHEME: WINNERS KILL LOOSERS
POPULATION
GOOD BAD
GOOD's
GOOD
BABIES

37. NEW SCHEME: WINNERS KILL LOOSERS
ONLY
POPULATION IF THEY
LOOK
LIKE
GOOD BAD THEM
GOOD's
GOOD
BABIES

38. NEW SCHEME: WINNERS KILL LOOSERS
ONLY
POPULATION IF THEY
LOOK
LIKE
GOOD BAD THEM
Don't kill koalas;
GOOD's their food is
GOOD useless to you.
BABIES They can be useful in a
future world.

39. CLEARING ALGORITHM
Input: population Output: smaller population
1) Sort the individuals (the best first)
2) Loop on individuals; for each of them (if alive)
kill individuals which are
(i) worse
(ii) at distance < some constant
(...but the  first murders are sometimes canceled)
3) Keep at most  individuals (the best)

40. I. Multimodal optimization
II. Adversarial / robust cases
a) What does it mean ?
b) Nash's stuff: computation
c) Robustness stuff: computation
III. Noisy cases

41. What is adversarial / robust optimization ?
I want to find argmin f.
But I don't know exactly f
I just know that f is in a family of functions
parameterized by .
I can just compute f(x,), for a given .

42. What is adversarial / robust optimization ?
I want to find argmin f.
But I don't know exactly f
I just know that f is in a family of functions parameterized by .
I can just compute f(x,), for a given .
Criteria:
Argmin E f(x,). (average case; requires a
X . probability distribution)
Or
Argmin sup f(x,). <== this is adversarial / robust.
X 

43. What is adversarial / robust optimization ?
I want to find argmin f.
But I don't know exactly f
We'll see
this case later.
I just know that f is in a family of functions parameterized by .
I can just compute f(x,), for a given .
Criteria:
Argmin E f(x,). (average case; requires a
X . probability distribution)
Or
Argmin sup f(x,). <== this is adversarial / robust.
X 
Let's see this.

44. ==> looks like a game; two opponents choosing their strategies.
VS
x y(=)
Can he try multiple times ?
Does he choose his strategy first ?

45. Let's have a high-level look at all this stuff.
We have seen two cases:
inf sup f(x,y) (best power plant for
x y worst earthquake ?)
inf E f(x,y) (best power plant for
x y average earthquake)
Does it really cover all possible models ?

46. Let's have a high-level look at all this stuff.
We have seen two cases: A bit pessimistic.
This is the performance if y is
inf sup f(x,y) chosen by an adversary who:
x y - knows us
- and has infinite computational resources
inf E f(x,y)
x y
Does it really cover all cases ?

47. Let's have a high-level look at all this stuff.
We have seen two cases: A bit pessimistic.
This is the performance if y is
inf sup f(x,y) chosen by an adversary who:
x y - knows us (or can do many trials)
- and has infinite computational resources
inf E f(x,y)
x y Good for nuclear
power plants.
Does it really cover all cases ?
Not good for economy,
games...

48. Let's have a high-level look at all this stuff.
We have seen two cases: A bit pessimistic.
This is the performance if y is
inf sup f(x,y) chosen by an adversary who:
x y - knows us
- and has infinite computational resources
inf E f(x,y)
x y Good for nuclear
power plants.
Does it really cover all cases ?
Not good for economy,
games...

49. Decision = positionning my distribution points.
Reward = size of Voronoi cells.
Newspapers distribution,
or Quick vs MacDonalds
(if customers ==> nearest)

50. Decision = positionning my distribution points.
Reward = size of Voronoi cells.
Simple model:
Each company makes his positionning
am morning for the whole day.
Newspapers distribution,
or Quick vs MacDonalds
(if customers ==> nearest)

51. Decision = positionning my distribution points.
Reward = size of Voronoi cells.
Simple model:
Each company makes his positionning
am morning for the whole day.
But choosing among strategies
(i.e. decision rules, programs)
leads to the same maths.
Newspapers distribution,
or Quick vs MacDonalds
(if customers ==> nearest)

52. Ok for power plants and earthquakes.
Inf sup f(x,y) But for economical choices ?
x y
Replaced by sup inf f(x,y) ?
y x

53. Inf sup f(x,y)
x y
Replaced by sup inf f(x,y) ?
y x
Means that we play second.
==> let's see “real” uncertainty techniques.

54. inf sup f(x,y) is not equal to sup inf f(x,y)
x y y x
...but with finite domains:
inf sup E f(x,y) is equal to sup inf f(x,y)
L(x) L(y) L(y) L(x)
Von
Neumann

55. inf sup f(x,y) is not equal to sup inf f(x,y)
x y y x
E.g.: Rock-paper-scissors
Inf sup = I win
Sup inf = I loose
And sup inf sup inf sup inf... ==> loop
But equality in
Chess, draughts, … (turn-based, full-information)

56. inf sup f(x,y) is not equal to sup inf f(x,y)
x y y x
...but with finite domains:
inf sup E f(x,y) is equal to sup inf f(x,y)
L(x) L(y) L(y) L(x)
...and is equal to inf sup E f(x,y)
x L(y)
==> The opponent chooses a randomized strategy
without knowing what we choose.

57. inf sup f(x,y) is not equal to sup inf f(x,y)
x y y x
...but with finite domains:
inf sup E f(x,y) is equal to sup inf f(x,y)
L(x) L(y) L(y) L(x)
...and is equal to inf sup E f(x,y)
x L(y)
==> The opponent chooses a randomized strategy
without knowing what we choose.
==> Good model for average performance against
non-informed opponents.

58. Let's summarize.
Nash: best distribution against worst
distribution = good model if opponent can't
know us, if criterion = average perf.
Best against worst: good in many games, and
good for robust industrial design
Be creative: inf sup E is often reasonnable
(e.g. sup on opponent, E on climate...)

59. Let's summarize.
Nash: best distribution against worst
distribution = good model if opponent can't
know us, if criterion = average perf.
Best against worst: good in many games, and
Nash:
good for robust industrial
Adversarial bidding (economy). design
Choices on markets.
Be creative: inf sup E is often reasonnable
Military strategy.
(e.g.
Games. sup on opponent, E on climate...)

60. Let's summarize.
Nash: best distribution against worst
distribution = good model if opponent can't
know us, if criterion = average perf.
Best against worst: good in many games, and
good for robust industrial design
In repeated games, you can
(try to)
Be creative: inf sup E is often reasonnable
predict your opponent's decisions.
(e.g. sup on opponent, E on climate...)
E.g.: Rock Paper Scissors.

61. Let's summarize.
Nash: best distribution against worst
distribution = good model if opponent can't
know us, if criterion = average perf.
Best against worst:“worst-case” analysis:
Remarks on this good in many games, and
good foryou're stronger (e.g. computer against
- if
robust industrial design
humans in chess), take into account human's weakness
Be creative: inf supearnis often reasonnable
- in Poker, you will E more money by playing against weak
Opponents
(e.g. sup on opponent, E on climate...)

62. inf sup f(x,y) is not equal to sup inf f(x,y)
x y y x
...but with finite domains:
inf sup E f(x,y) is equal to sup inf f(x,y)
L(x) L(y) L(y) L(x)
...and is equal to inf sup E f(x,y)
x L(y)
L(x),L(y) is a
Nash equilibrium;
L(x), L(y) not necessarily unique.

63. I. Multimodal optimization
II. Adversarial / robust cases
a) What does it mean ?
b) Nash's stuff: computation
- fictitious play
- EXP3
- coevolution
c) Robustness stuff: computation
III. Noisy cases

64. FIRST CASE: everything finite ( x and  are in finite domains).
==> Fictitious Play (Brown, Robinson)
Simple,
consistant,
but slow.
(see however
Kummer's
paper)

65. FIRST CASE: everything finite ( x and  are in finite domains).
==> Fictitious Play (Brown, Robinson)
==> sequence of simulated games as follows:
(x(n),y(n)) defined as follows:
Simple,
consistant,
x(n+1) = argmin sum f(x, y(i)) but slow.
x i<n+1 (see however
y(n+1) = argmax sum f(x(i), y)
Kummer's
x i<n+1
paper)

66. FIRST CASE: everything finite ( x and  are in finite domains).
==> Fictitious Play (Brown, Robinson)
==> sequence of simulated games as follows:
(x(n),y(n)) defined as follows:
Simple,
consistant,
x(n+1) = argmin sum f(x, y(i)) but slow.
x i<n+1 (see however
y(n+1) = argmax sum f(x(i), y)
Kummer's
x i<n+1
paper)
==> … until n=N; play randomly one of the x(i)'s.

67. Much faster: EXP3, or INF (more complicated, see
Audibert)

68. Complexity of EXP3 within logarithmic terms:
K / epsilon2
for reaching precision epsilon.

69. Complexity of EXP3 within logarithmic terms:
K / epsilon2
for reaching precision epsilon.
==> we don't even read all the matrice of
2
the f(x,y) ! (of size K )

70. Complexity of EXP3 within logarithmic terms:
K / epsilon2
for reaching precision epsilon.
==> we don't even read all the matrice of
2
the f(x,y) ! (of size K )
==> provably better than all deterministic
algorithms

71. Complexity of EXP3 within logarithmic terms:
2
K / epsilon
for reaching precision epsilon.
==> we don't even read all the matrice of
2
the f(x,y) ! (of size K )
==> provably better than all deterministic
algorithms
==> requires randomness + RAM

72. Complexity of EXP3 within logarithmic terms:
K / epsilon2
for reaching precision epsilon.
Ok, it's great, but complexity K is still far too big.
For Chess, log10(K) l 43.
For the game of Go, log10(K) l 170.

73. Coevolution for K huge:
Population 1 Population 2

74. Coevolution for K huge:
Population 1 Population 2
(play games)

75. Coevolution for K huge:
Population 1 Population 2
Fitness = average
fitness against
opponents

76. Coevolution for K huge:
Population 1 Population 2
Fitness = average
Pop. 1 Pop. 2 fitness against
opponents

77. Coevolution for K huge:
Population 1 Population 2
Fitness = average
+ babies Pop. 1 Pop. 2 fitness against
opponents
Population 1 Population 2

78. RED QUEEN EFFECT in finite population:
(FP keeps all the memory, not coevolution)
1. Rock vs Scissor
2. Rock vs Paper
3. Scissor vs Paper
4. Scissor vs Rock
5. Paper vs Rock
6. Paper vs Scissor
7. Rock vs Scissor =1

79. RED QUEEN EFFECT:
1. Rock vs Scissor
2. Rock vs Paper
....
6. Paper vs Scissor
7. Rock vs Scissor =1
==> Population with archive (e.g.
the best of each past iteration – looks like FP)

80. Reasonnably good in very hard cases
but plenty of more specialized algorithms for
various cases:
- alpha-beta
- (fitted)-Q-learning
- TD(lambda)
- MCTS
- ...

81. I. Multimodal optimization
II. Adversarial / robust cases
a) What does it mean ?
b) Nash's stuff: computation
c) Robustness stuff: computation
III. Noisy cases

82. c) Robustness stuff: computation
==> we look for argmin sup f(x,y)
x y
==> classical optimization for x
==> classical optimization for y with
restart

83. Naive solution for robust optimization
Procedure Fitness(x)
y=maximize f(x,.)
Return f(x,y)
Mimimize fitness

84. Warm start for robust optimization
==> much faster
Procedure Fitness(x)
Static y=y0;
y = maximize f(x,y)
Return f(x,y)
Mimimize fitness

85. Population-based warm start for
robust optimization
Procedure Fitness(x)
Static P=P0;
P = multimodal-maximize f(x,P)
Return max f(x,y)
y in P
Mimimize fitness

86. Population-based warm start for
robust optimization
Procedure Fitness(x)
f(x,y) = sin( y ) + cos((x+y) / pi) x,y in [-10,10]
Static P = multimodal-maximize f(x,P)
==> global maxima (in y) moves quickly (==> looks hard)
Return max f(x,y)
==> but local maxima (in y) move slowly as a function of x
y in P
==> warm start very efficient
Mimimize fitness

87. Population-based warm start for
robust optimization
Procedure + cos((x+y) / pi)
f(x,y) = sin( y )
Fitness(x) x,y in [-10,10]
Static P = multimodal-maximize f(x,P)
==> global maxima (in y) moves quickly (==> looks hard)
Return max f(x,y)
==> but local maxima (in y) move slowly as a function of x
==> warm start very efficient
y in P
==> interestingly, we just rediscovered coevolution :-)
Mimimize fitness

89. Plots of f(x,.)
for various values of x.

90. Coevolution for Nash:
Population 1 Population 2
Fitness = average Fitness = average
fitness against fitness against
Pop. 1 Pop. 2
opponents opponents

91. Coevolution for inf sup:
Population 1 Population 2
Fitness = worst Fitness = average
fitness against fitness against
Pop. 1 Pop. 2
opponents opponents

92. I. Multimodal optimization
II. Adversarial / robust cases
III. Noisy cases

93. Arg min E f(x,y)
x y
1) Monte-Carlo estimates
2) Races: choosing the precision
3) Using Machine Learning

94. Monte-Carlo estimates
E f(x,y) = 1/n sum f(x, yi) + error
Error scaling as 1/squareRoot( n )
==> how to be faster ?
==> how to choose n ? (later)

95. Faster Monte-Carlo evaluation:
- more points in critical areas (prop. to
standard deviation), (+rescaling!)
- evaluating f-g, where g has known
expectation and is “close” to f.
- evaluating (f + f o h)/2
where h is some symmetry

96. Faster Monte-Carlo evaluation:
- more points in critical areas (prop. to
standard deviation), (+rescaling!)
- evaluating f-g, where g has known
expectation and is “close” to f.
Importance sampling
- evaluating (f + f o h)/2
where h is some symmetry

97. Faster Monte-Carlo evaluation:
- more points in criticalControl variable to
areas (prop.
standard deviation), (+rescaling!)
- evaluating f-g, where g has known
expectation and is “close” to f.
- evaluating (f + f o h)/2
where h is some symmetry

98. Faster Monte-Carlo evaluation:
- more points in critical areas (prop. to
standard deviation), (+rescaling!)
- evaluating f-g, where g has variables
Antithetic known
expectation and is “close” to f.
- evaluating (f + f o h)/2
where h is some symmetry

99. And quasi-Monte-Carlo evaluation ?
d
log(n) / n instead of std / sqrt(n)
==> much better for n really large (if some smoothness)
==> much worse for d large

100. Arg min E f(x,y)
1) Monte-Carlo estimates
2) Races: choosing the precision
3) Using Machine Learning

101. Bernstein's inequality (see Audibert et al):
Where:
R = range of the Xi's

102. Comparison-based optimization with noise:
how to find the best individuals by a race ?
While (I want to go on)
{
I evaluate once each arm.
I compute a lower and upper bound for all
non-discarded individuals
(by Bernstein's bound).
I discard individuals which are excluded
by the bounds.
}

103. Trouble:
What if two individuals have the same fitness value ?
==> infinite run !
Tricks for avoiding that at iteration N:
- limiting number of evals by some function of N
(e.g. exp(N) if a log-linear convergence is
expected w.r.t number of iterations)
- limiting precision by some function of sigma
(e.g. precision = sqrt(sigma) / log(1/sigma) )

104. Arg min E f(x,y)
1) Monte-Carlo estimates
2) Races: choosing the precision
3) Using Machine Learning

105. USING MACHINE LEARNING
Statistical tools: f ' (x) = approximation
( x, x1,f(x1), x2,f(x2), … , xn,f(xn))
y(n+1) = f ' (x(n+1) )
e.g. f' = quadratic function closest to f on the x(i)'s.
and x(n+1) = random or x(n+1) = maxUncertainty
or x(n+1) = argmin Ef or x(n+1)=argmax E max(0,bestSoFar - Ef(x))

106. USING MACHINE LEARNING SVM,
Kriging, RBF,
Gaussian Processes,
Or f'(x) = maximum likelihood
on a parametric noisy model
Statistical tools: f ' (x) = parametric approximation
( x, x1,f(x1), x2,f(x2), … , xn,f(xn))
y(n+1) = f ' (x(n+1) )
e.g. f' = logistic quadratic function closest to f on the x(i)'s.
and x(n+1) = random or x(n+1) = maxUncertaintyRed
or x(n+1) = argmin f' or x(n+1)=argmax E max(0,bestSoFar - f(y))

107. USING MACHINE LEARNING
Very fast if good model
Statistical tools: f ' (x) = approximation model + noise)
(e.g. quadratic
Expensive but efficient: x, x1,f(x1), x2,f(x2), … , xn,f(xn))
(
Efficient Global f ' (x(n+1) )
y(n+1) =
Optimization
e.g. f' = quadratic function closest to f on the x(i)'s.
in multimodal context
and x(n+1) = random or x(n+1) = maxUncertaintyRed
or x(n+1) = argmin f' or x(n+1)=argmax E max(0,bestSoFar - f(y))

108. USING MACHINE LEARNING
Very fast if good model
Statistical tools: f ' (x) = approximation model + noise)
(e.g. quadratic
( x, x1,f(x1), x2,f(x2), … , xn,f(xn))
Alsoy(n+1) = f ' (x(n+1) )
termed
“expected improvement”
e.g. f' = quadratic function closest to f on the x(i)'s.
and x(n+1) = random or x(n+1) = maxUncertaintyRed
or x(n+1) = argmin f' or x(n+1)=argmax E max(0,bestSoFar - Ef(x))

109. So some tools for noisy optimization:
1. introducing races + tricks
2. using models equipped
with uncertainty
(2. more difficult, but
sometimes much more efficient)

110. Deng et al: application to DIRECT

111. A rectangle j or size alpha(j) is splitted if for some K:

112. A rectangle j or size alpha(j) is splitted if for some K:

113. Warm starts Robust optimization
Coevolution Noisy optimization
Niching Nash equilibria
Surrogate models
Clearing
Maximum uncertainty
Multimodal optimization
Monte-Carlo estimate
Restarts (sequential / parallel) Quasi-Monte-Carlo
Koalas and Eucalyptus Van Der Corput
Dinosaurs (and other mass extinction) Halton
Fictitious Play Scrambling
EXP3 Antithetic variables, control
EGO variables, importance sampling.

114. EXERCISES (to be done at home, deadline given soon):
1. Implement a (1+1)-ES and test it on
the sphere function f(x)=||x||^2 in
dimension 3. Plot the convergence.
Nb: choose a scale so that you can check
the convergence rate conveniently.
2. Implement a pattern search method (PSM)
and test it on the sphere function, also in
dimension 3 (choose the same scale as above).

115. 3. We will now consider a varying dimension;
- x(n,d) will be the nth visited point in dimension d for the (1+1)-ES, and
- x'(n,d) will be the nth visited point in dimension d for the PSM.
For both implementations, choose e.g. n=5000, and plot x(n,d) and
x'(n,d) as a function of d. Choose a convenient scaling as a function of d
so that you can see the effect of the dimension.
4. What happens if you consider a quadratic positive definite
ill-conditionned function instead of the sphere function ?
E.g.: f(x) = sum of 10^i xi^2
= 10x(1)^2 + 100x(2)^2 + 1000x(3)^2 in dimension 3; test in
Dimension 50.
5. (without implementing) Suggest a solution for the problem in Ex. 4.