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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Multi-Objective Optimal Control

Massimiliano Vasile Aerospace Centre of Excellence, Department of Mechanical and Aerospace Engineering University of Strathclyde, Glasgow (United Kingdom)

September 1, 2017

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Pareto Dominance and Efficiency

Pareto Dominance Consider the vector functions F : Rn → Rm, with F(x) = [f1(x), f2(x), ..., fi(x), ..., fm(x)]T , g : Rn → Rq, with g(x) = [g1(x), g2(x), ..., gj(x), ..., gq(x)]T and problem min

x

F s.t. (MOP) g(x) ≤ 0 Given the feasible set X = {x ∈ Rn|g(x) ≤ 0} and two feasible vectors x, ˆ x ∈ x, we say that x is dominated by ˆ x if fi(ˆ x) ≤ fi(x) for all i = 1, ..., m and there exists a k so that fk(ˆ x) = fk(x). We use the relation ˆ x ≺ x that states that ˆ x dominates x. Pareto Efficiency A vector x∗ ∈ X will be said to be Pareto efficient, or optimal, with respect to Problem (MOP) if there is no other vector x ∈ X dominating x∗ or: x ⊀ x∗, ∀x ∈ X Pareto Set All non-dominated decision vectors in X form the Pareto set XP and the corresponding image in criteria space is the Pareto front.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Karush-Khun-Tucker Optimality Conditions

[Cha08]

Necessary condition for x∗ to be locally optimal.

Theorem (KKT)

If x∗ ∈ X is an efficient solution to problem MOP, then there exist vectors η ∈ Rm and λ ∈ Rq such that:

m

• i

ηi∇fi(x∗) +

q

• j

λj∇gj(x∗) = 0 (1) gj(x∗) = 0, j = 1, ..., q (2) λj ≥ 0, j = 1, ..., q (3) ηi ≥ 0, i = 1, ..., m (4) ∃ηi > 0 (5)

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Pareto Set and Front

In the unconstrained case KKT optimality conditions reduce to:

m

• i

ηi∇fi(x∗) = 0 (6) ηi ≥ 0, i = 1, ..., m (7) ∃ηi > 0 (8) Condition 6 leads to an interesting result (Hillermeier2001 [Hil01]) that the Pareto set is an m − 1 dimensional manifold. This also implies that the Pareto set has zero measure in Rn with m ≤ n.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Multi-Objective Optimal Control

Consider the following multi-objective optimal control problem (MOCP): min

u F

s.t. (MOCP) ˙ x = h(x, u, p, t) g(x, u, p, t) ≤ 0 ψ(x0, xf , t0, tf ) ≤ 0 t ∈ [t0, tf ] where F is a vector function of the state variables x : [t0, tf ] → Rn, control variables u ∈ L∞, time t and some static parameters p ∈ Rq. Functions x belong to the Sobolev space W 1,∞, objective functions are fi : Rn+2n × Rp × [t0, tf ] − → R, h : Rn × Rp × Rq × [t0, tf ] − → Rn, algebraic constraints g : Rn × Rp × Rq × [t0, tf ] − → Rs, and boundary conditions R2n+2 − → Rq. Note that problem (MOCP) is generally non-smooth and can have many local minima.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

MOCP: How to Solve it?

• Option 1 is to attempt the solution of the problem in

vector form.

• Option 2 is to find a suitable form of scalarisation and

then use the existing machineries to solve single objective

• ptimal control problems.
• Option 3 is to use a mix of Option 1 and Option 2.

In the following we will introduce some suitable scalarisation techniques and we will then show how to combine Option 1 and Option 2 into a single method with some desirable theoretical and computational properties.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Pascoletti-Serafini Scalarisation[Eic08]

The scalarisation of Pascoletti-Serafini is based on the idea of descent cones K. An optimal (K-minimal) solution to problem MOP is solution to the following constrained single objective optimisation problem: mint t s.t. at − F(x) + r ∈ K g(x) ≤ 0 (9)

• r, in a more computationally friendly, form:

mins s s.t. wj(fj(x) − zj) ≤ s ∀j = 1, ..., m g(x) ≤ 0 (PS) A point is K-minimal when: (¯ F − K) ∩ F(X) = {¯ F} From this definition one can understand that a K-minimal point is Pareto efficient.

K K f1 f2 F=[f1,f2]

F(X)

r+at

F(X)

K f1 f2 F=[f1,f2] 7 / 38

UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Chebyshev Scalarisation[Eic08]

Chebyshev scalarisation is based on the idea of descent directions identified by the weights w: minx∈X maxj∈{1,...,m} wj(fj(x) − zj) s.t. g(x) ≤ 0 (CS)

Theorem (CS)

A point (s, x) ∈ R × X is a minimal solution

• f problem (PS) with z ∈ Rm,

zj < minx∈X fj(u), j = 1, ..., m, and w ∈ int(Rm

+), if and only if x is a solution of

problem (CS). From theorem CS one can expect that the solution of the PS and CS problems are

• equivalent. This is an important property

when designing algorithms because, in some cases, the solution of PS translates into the solution of CS.

f1 f2 F(x)

Increasing g(F) Decreasing g(F)



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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

(Scalar) Pontryagin Maximum Principle

Given the following optimal control problem in Mayer’s form: min f (xf , tf ) s.t ˙ x = h(x, u, t) g(x, u, t) ≥ 0 ψ(x0, xf , t0, tf ) ≥ 0 t ∈ [t0, tf ] If u∗ is a locally optimal solution for problem (PSOCP) then there exist a vector η ∈ Rm, λ ∈ Rn and a vector µ ∈ Rq such that: u∗ = argmin

u∈U

(λTh(x∗, u, t) + µTg(x∗, u, t)) λT∇xh(x∗, u∗, t) + µT∇xg(x∗, u∗, t) + ˙ λ = 0 λ ≥ 0; µ ≥ 0 with transversality conditions: ∇xf + νT∇xψ = λx(tf ) ν ≥ 0

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Pascoletti-Serafini Scalarised MOCP

Consider each objective function to be fj(xf , tf ) and the scalarised Multi-Objective Optimal Control problem: minsf sf s.t. wj(fj(xf , tf ) − zj) − sf ≤ 0 ∀j = 1, ..., m ˙ x = h(x, u, t) g(x, u, t) ≥ 0 ψ(x0, xf , t0, tf ) ≥ 0 t ∈ [t0, tf ] (PSOCP) If s is a slack variable with final condition sf and zero time variation ˙ s = 0, then problem (PSOCP) presents itself in a form similar to Mayer’s

• problem. The major difference is the mixed boundary constraint on xf , tf

and sf for every j = 1, ..., m.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Necessary Conditions for Local Optimality

Theorem (Vasile2017)

Consider the function H = λTh(x, u, t) + µTg(x, u, t). If u∗ is a locally

• ptimal solution for problem (PSOCP), with associated state vector x∗,

and H is Frechet differentiable at u∗, then there exist a vector η ∈ Rm, λ ∈ Rn and a vector µ ∈ Rq such that: u∗ = argmin

u∈U

λTh(x∗, u, t) + µTg(x∗, u, t) λT∇xh(x∗, u∗, t) + µT∇xg(x∗, u∗, t) + ˙ λ = 0 ˙ λs = 0 λ ≥ 0; µ ≥ 0 with transversality conditions: 1 −

m

• j

ηj = λs(tf ) ηT∇xF + νT∇xψ = λx(tf ) η > 0; ν ≥ 0

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Example

Consider the very simple one-dimensional controlled dynamical system with constant control acceleration and two objectives on the terminal states: min sf − w1xf < sf w2vf < sf ˙ x = v; ˙ v = −u; ˙ s = 0; ˙ λs = 0; ˙ λx = 0; ˙ λv = −λx; x(t0) = 0; v(t0) = 1; 0 ≤ u ≤ 1 sf ≥ 0 with xf = x(tf ), vf = v(tf ), sf = s(tf ) and terminal conditions: λs(tf ) = 1 − η1 − η2; λx(tf ) = −η1w1; λv(tf ) = η2w2;

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Example

The solution of the controlled dynamics is given by: x = t2 2 + t t ∈ [t0, t1] v = t + 1 t ∈ [t0, t1] x = vf t + x1 t ∈ [t1, tf ] v = v1 = vf t ∈ [t1, tf ] x1 = x(t1); v1 = v(t1) In this case it is easy to demonstrate that the Pareto front is given by the following second order algebraic equation: xf = 1 + 2vf − v 2

f

2 We want to show that all the points along the front satisfy the optimality conditions and represent a minimum for sf .

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Example

Consider first the extreme values: minsf − xf ≤ sf x = t2 2 + t t ∈ [t0, t1] v = t + 1 t ∈ [t0, t1] x = vf t + x1 t ∈ [t1, tf ] v = v1 = vf t ∈ [t1, tf ] x1 = x(t1); v1 = v(t1)

vf

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• xf
• 1
• 0.95
• 0.9
• 0.85
• 0.8
• 0.75
• 0.7
• 0.65
• 0.6
• 0.55
• 0.5

By imposing the continuity conditions at t1 we get a simple algebraic problem: minsf x1 = −t2

1

2 + t1 vf = −t1 + 1 − sf = vf tf + x1 minsf − sf = 1 − t2

1

2 0 ≤ t1 ≤ 1 sf = −1; t1 = 0 sf = −1/2; t1 = 1

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Example

We now need to verify that we can find a suitable set of Lagrange multipliers that satisfy the necessary conditions: H = λxv − λvu + µ1(u − 1) − µ2u ∂H ∂u = −λv + µ1 − µ2 λv = −λx(t − tf ) + λv(tf ) λv(tf ) = 0 λx(tf ) = −η1 λv < 0 ∀t ∈ [t0, tf ] These equations confirm that there is a single switching point for the control u∗. The conditions on the multipliers associated to the slack variable sf are always satisfied.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Transcription of the MOCP

We now have a scalarised and a vector from of the MOCP. In both cases the formulation contains a mix of algebraic and differential equations (DAE). The next step is to transcribe the infinite dimensional system of DAE into a finite dimensional Nonlinear Programming Problem that can be solved numerically. The transcription technique proposed here is based on Finite Elements in time on spectral basis.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Direct Finite Element Transcription

(Vasile 2000, 2003, 2010, [VF00][VBZ03][Vas10])

t x,u gap{ gap{ gap{ }gap tf t0

Transcription of the optimal control problem (PSOCP) into a finite dimensional Nonlinear Programming Problem. We start from decomposing the time domain in a finite number of time elements.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Bi-discontinuos Integral Form

The differential constraints are recast in weak form and integrated by parts, leading to: tf

t0

˙ wTx + wTh(x, u, t)dt − wT

f xb f + wT 0 xb 0 = 0

(10) where w are generalised weight functions and xb are the boundary values of the states, that may be either imposed or free.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

State and Control Transcription

Given the partition of the time domain: D =

N

• j=1

Dj(tj−1, tj) (11)

• ne can parametrise, over each Dj, the states, controls and weight

functions as x(t) =

N

℧

j=1Xj = N

℧

j=1 l

• s=0

φsj(t)xsj (12) u(t) =

N

℧

j=1Uj = N

℧

j=1 m

• s=0

γsj(t)usj (13) w(t) =

N

℧

j=1Wj = N

℧

j=1 l+1

• s=0

θsj(t)wsj (14) where

N

℧

j=1 denotes the juxtaposition of the polynomials defined over each

sub-interval, φsj(t), γsj(t) and θsj(t) indicate the s-th polynomial over element j and are chosen among the space of polynomials of degree l, m and l + 1 respectively, while xsj, usj and wsj denote the nodal values of the states, control and test functions.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

State and Control Transcription (2)

It is practical to define each Dj over the normalised interval [−1, 1] through the transformation τ = 2t − tj−tj−1

2

tj − tj−1 (15) This way it’s easy to express the polynomials φsj(t), γsj(t) and θsj(t) as the Lagrange interpolation on the Gauss nodes in the normalised interval: φsj(t) = φsj(τ) =

l

• k=0,k=s

τ − τk τs − τk (16) where τ∗ indicates a Gauss node, and similarly can be done for γsj and θsj. Different Gauss nodes will lead to schemes with slightly different characteristics.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Transcribed Problem

(Vasile 2000, 2003, 2010, [VF00][VBZ03][Vas10])

˜ fi =αifi

• Xb

0, Xb f , Ub 0, Ub f , t0, tf

• +

βi

N

• j=1

l+1

• k=1

σkLi (Xj(τk), Uj(τk), τk) ∆tj 2 (17) and for the variational constraints leads for each element j to the system cj =

l+1

• k=1

σk

• ˙

Wj(τk)TXj(τk) + Wj(τk)Thj(τk)∆tj 2

• − WT

p+1,jXb j + WT 1,jXb j−1 = 0

(18)

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

DFET Transcribed PSOCP

Vasile and Ricciardi 2016[VR16]

Once the differential equations have been transcribed into a set

• f nonlinear algebraic equations, the original optimal control

problem can be casted in the Pascoletti-Serafini form and solved with a standard NLP solver: minsf sf s.t. wj(fj(Y, t) − zj) − sf ≤ 0 ∀j = 1, ..., m c(Y, t) ≥ 0 t ∈ [t0, tf ] (PSDFET) where Y = [X, U, Xb

0, Xb f , t0, tf ] is the decision vector of the

NLP problem and boundary conditions ψ(Xb

0, Xb f , t0, tf ) ≥ 0

are included in the constraint vector c.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Necessary Optimality Conditions of the Transcribed Problem

Theorem

If Y ∗ is a locally optimal solution for problem (PSDFET), then there exist a vector η ∈ Rm, λ ∈ Rn such that: λT∇Y c(Y, t) + ηT∇Y wTF(Y, t) = 0 1 −

• ηj = 0

λ ≥ 0; η > 0 From the definition of Pascoletti-Serafini scalarisation we know that a K-efficient solution of problem (PSDFET) is locally optimal and satisfies the above theorem. To be noted that this theorem is equivalent to the KTT conditions previously defined. In the following we will distinguish between the

• ptimisable parameters p = [U, Xb

0, Xb f , t0, tf ]T and the state parameters

x = X.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Solving the Transcribed MOCP

The transcribed PSOCP suggests that a standard NLP solver can be used to find a Pareto efficient solution. A direct application of this technique will provide a single point

• n the Pareto front unless a strategy is implemented to change

the weight vector w (see for example the Normal Boundary Intersection strategy). Alternatively we can use a method devised to solve vector MOO problems and generate a population of solutions each associated to a different w. We can then use the PSOCP to locally converge to a Pareto efficient point from each solution in the population. In the following we will explain how.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Multi-Agent Collaborative Search (1)

[Vas][VZ11][ZV13][RV15]

Seed agents (Latin Hypercube) J1 J2 Associate social agents to a weight vector J1 J2 Update archive with non dominated solutions J1 J2 Update archive with non dominated solutions J1 J2 Perform social actions (DE with agents or archive) J1 J2 Repeat until feval < max Perform individual actions (inertia, pattern search, DE) J1 J2 25 / 38

UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Multi-Agent Collaborative Search (2)

During the exploration phase no gradient information is used and the agents converge, in parallel, towards the Pareto set using a series of sampling heuristics. From Theorem 2 we know that the solution of problem (PS) is also solution of problem (CS). Therefore, when agents do not implement any gradient-based local search approach, they solve the following problem: min

p∈X max j

wj(fj − zj) (19) The assumption is that the control vector p is in the feasible set, or, in other words, that all constraints are satisfied.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Bi-level Formulation

Vasile and Ricciardi 2016 - [RVM16][RVTM16][VR16]

In order to maintain feasibility, problem (19) is solved with the following bi-level formulation: minpc∈X maxj wj(fj(x∗, p∗) − zj) s.t. (x∗, p∗) = argminp{(p − pc)T(p − pc)|c(x, p) ≥ 0} fj(x∗, p∗) =

• fj(x∗, p∗)

if c(x∗, p∗) ≥ 0 L + knf

• therwise

(20) Once an agent decides to trigger a local search with a gradient method an NLP solver is invoked and directly applied to problem (PSDFET).

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Goddard Rocket - (Ricciardi and Vasile 2016)

Physical model and constraints

• Constant gravity acceleration g
• Constant thrust acceleration a
• Control parameters: thrust

angle u

• At final time, altitude must be

h

• At final time, vertical velocity

must be 0

Numerical settings

• 10000 fun evals, 10 agents
• 10 solutions in the archive
• 160 variables

g a u x y h

Objectives

• Minimise mission time
• Maximise horizontal velocity

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Goddard Rocket

Ricciardi and Vasile 2016 - [RVM16] min

tf ,u [f1, f2]T = [tf , −vx(tf )]T

(21)          ˙ x =vx ˙ vx =a cos u ˙ y =vy ˙ vy = − g + a sin u      x(0) = 0; vx(0) = 0 y(0) = 0; vy(0) = 0 y(tf ) = h; vy(tf ) = 0

x[LU]

• 20

20 40 60 80 100 120 y[LU]

• 2

2 4 6 8 10 12 Trajectory 1 Trajectory 2 Trajectory 3 Trajectory 4

Trajectories

tf[TU] 100 150 200 250

• vx(tf)[LU/TU]
• 1
• 0.9
• 0.8
• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

1 2 3 4

Pareto front

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Goddard Rocket - (Ricciardi and Vasile 2016)

t[TU] 50 100 150 200 250 u[rad]

• 3
• 2
• 1

1 2 3 u u(analytic)

Control law for point 1

t[TU] 50 100 150 200 250 u[rad]

• 3
• 2
• 1

1 2 3 u u(analytic)

Control law for point 2

t[TU] 50 100 150 200 250 u[rad]

• 3
• 2
• 1

1 2 3 u u(analytic)

Control law for point 3

t[TU] 50 100 150 200 250 u[rad]

• 3
• 2
• 1

1 2 3 u u(analytic)

Control law for point 4

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Optimal Descent - (Ricciardi and Vasile 2017)

Multi-objective version of a problem proposed in [BK02]. min[−θ(tf ), qU]T s.t. ˙ h = v sin γ ˙ φ = v r cos γ sin ψ/ cos θ ˙ θ = v r cos γ cos ψ ˙ v = −D m − g sin γ ˙ γ = L mv cos β + cos γ v r − g v

• ˙

ψ = 1 mv cos γ L sin β + v r cos θ cos γ sin ψ sin θ q ≤ qU

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Optimal Descent - (Ricciardi and Vasile 2017)

The extreme values correspond to the single objective solutions in [BK02].

• 0.55
• 0.5
• 0.45
• 0.4
• 0.35
• 0.3
• 0.25
• f [rad]

20 25 30 35 40 45 50 55 60 65 70 75 q peak [btu/ft 2/s] 1 2 3 4 5 6 7 8 9 10 500 1000 1500 2000 2500 time [s] 0.6 0.8 1 1.2 1.4 1.6 1.8 2 L/D [rad/s] Solution 1 Solution 2 Solution 3 Solution 4 Solution 5 Solution 6 Solution 7 Solution 8 Solution 9 Solution 10

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Optimal Descent - (Ricciardi and Vasile 2017)

500 1000 1500 2000 2500 time [s] 10 20 30 40 50 60 70 80 [deg] Solution 1 Solution 2 Solution 3 Solution 4 Solution 5 Solution 6 Solution 7 Solution 8 Solution 9 Solution 10 500 1000 1500 2000 2500 time [s] 10 20 30 40 50 60 70 q [btu/ft 2/s] Solution 1 Solution 2 Solution 3 Solution 4 Solution 5 Solution 6 Solution 7 Solution 8 Solution 9 Solution 10 500 1000 1500 2000 2500 time [s] 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 altitude [ft] 10 5 Solution 1 Solution 2 Solution 3 Solution 4 Solution 5 Solution 6 Solution 7 Solution 8 Solution 9 Solution 10 5 x 10 6 10 2 4 0.5 z 10 6 6 10 7 y 15 1 8 10 1.5 2 Solution 1 Solution 2 Solution 3 Solution 4 Solution 5 Solution 6 Solution 7 Solution 8 Solution 9 Solution 10

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

Final Remarks I

• A multi-objective optimal control problem can be

reformulated with an appropriate scalarisation approach into a constrained single objective problem.

• Necessary conditions for optimality were derived for the

scalarised MOCP.

• The scalarisation technique called Pascoletti-Serafini is

equivalent to a weighted Chebyshev scalerisation.

• The transcribed PSOCP can be solved with a memetic

algorithm providing globally efficient solutions.

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References I

JT Betts and I Kolmanovsky, Practical methods for

• ptimal control using nonlinear programming, Applied

Mechanics Reviews 55 (2002), B68.

• Y. Yacov Chankong, Vitra; Haimes, Multiobjective decision

making, Dover Publications, Inc., 2008. Gabriele Eichfelder, Adaptive scalarization methods in multiobjective optimization, Springer-Verlag Berlin Heidelberg, 2008, DOI 10.1007/978-3-540-79159-1, 2008. Claus Hillermeier, Nonlinear multiobjective optimization, International Series of Numerical Mathematics, Birkhuser Basel.DOI 10.1007/978-3-0348-8280-4., 2001. Lorenzo Angelo Ricciardi and Massimiliano Vasile, Improved archiving and search strategies for multi agent collaborative search, EUROGEN2015, 2015.

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UTOPIAE Definitions and Preliminary Ideas Scalarisation Methods Necessary Conditions for Optimality Direct Finite Element Transcription Solution with Memetic Algorithms Examples of Application Final Remarks References

References II

Lorenzo A Ricciardi, Massimiliano Vasile, and Christie Maddock, Global solution of multi-objective optimal control problems with multi agent collaborative search and direct finite elements transcription, Evolutionary Computation (CEC), 2016 IEEE Congress on, IEEE, 2016,

• pp. 869–876.

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