Weighted Sum Method

The idea of the weighted sum method is to provide the different objective functions with positive weights and then to minimize the sum of the weighted objective functions. This is probably the most intuitive way to couple the objective functions as one can directly specify how important it is to lower the i-th objective function fi in comparison to the other objective functions by providing the ratio of the weights.

Abstractly we consider for some weightsα∈R^k the weighted sum problem

u∈Umin_ad

k

X

i=1

αifi(u). (WSP)

Notation 3.13. For α∈R^k we denote the weighted sum problem with weights αby (WSP)_α. Furthermore, we define F_α :U_ad →R,F_α(u) :=Pk

i=1α_if_i(u) for all u∈U_ad. Analytical Results

Lemma 3.14. Let Assumption 1 be satisfied and α∈R^k. Then (WSP)α has a unique Pareto optimal solution.

Proof. It is easy to check that the functiongα:R^k_≥yid →R, gα(x) :=Pk

i=1αixi fulfils Presump-tion 1, if α∈R^k holds.

Remark 3.15. Having the notation of the introduction to this section in mind, we consider the family of functions (gα)_α∈R^k withgα:R^k_≥yid →R, gα(x) :=Pk

i=1αixi for all α∈R^k.

Corollary 3.16. Let Assumptions 1 and 2 be satisfied. Let furthermore α ∈ R^k be arbitrary.

Then the first-order condition stated in Theorem 3.11 for the unique solutionu¯ of (WSP)_α reads h

k

X

i=1

α_i∇f_i(¯u), u−ui¯ _U ≥0 for allu∈U_ad. (3.2) In a next step we want to investigate which points on the Pareto front can be obtained by solving a weighted sum problem. The main result will be that all Pareto optimal points can be computed by a weighted sum problem, if Assumption 1 is satisfied. To be able to show this result, we first have to establish some small statements.

Lemma 3.17. Let X ⊂ R^k be arbitrary. Then PX = P_X+R^k

≥, i.e. the set of Pareto optimal points does not change when adding R^k≥ to the set X.

Proof. Let x∈P_X be arbitrary. So there is no ˜x ∈X with ˜xx. It follows immediately that there is no ˆx∈X+R^k≥ with ˆxx.

Now let reverselyz∈P_X+R^k

≥ be arbitrary, i.e. z=x+ywithx∈Xandy ∈R^k_≥. Asz is Pareto optimal, it is clear that y = 0 has to hold. But then z ∈ X and because ofX ⊂X+R^k≥ and the Pareto optimality ofz inX+R^k_≥, we obtain thatz∈PX.

Definition 3.18. A setX ⊂R^k is called R^k≥-convex, if the set X+R^k≥ is convex.

Lemma 3.19. Let f1, . . . , f_k be convex and Y :=f(U_ad). Then Y is R^k≥-convex.

Proof. Lety1, y2 ∈Y+R^k_≥andλ∈(0,1) be arbitrary. So there areu1, u2 ∈Uadandx1, x2∈R^k_≥ withyi=f(ui)+xifori= 1,2. Then we obtain by using the convexity of the functionsf1, . . . , f_k

λy₁+ (1−λ)y₂ =λf(u₁) + (1−λ)f(u₂) +λx₁+ (1−λ)x₂

≥f(λu1+ (1−λ)u2) +λx1+ (1−λ)x2. Withv:=λu₁+ (1−λ)u₂ ∈U_ad we get

λy₁+ (1−λ)y₂=f(v) + ˜x, for a ˜x∈R^k≥, so thatλy1+ (1−λ)y2 ∈Y +R^k≥.

The next theorem is the main theorem of this section. It states that all Pareto optimal points can be obtained by solving a weighted sum problem. The statement and the main idea of the proof are taken from [11, Theorem 3.5].

Theorem 3.20. Let Assumption 1 be satisfied. If y¯=f(¯u) ∈ P_Y, then there is α ∈R^k such thatu¯ is the unique solution of (WSP)α.

Proof. Let ¯y=f(¯u)∈PY be arbitrary. Then by Lemma 3.17 we get ¯y∈P_Y+R^k

≥ and thus

Y +R^k≥−y¯

∩R^k< =∅. (3.3)

According to Lemma 3.19 the set Y +R^k≥ is convex and consequentlyY +R^k≥−y¯is convex as well. SinceR^k_<is a convex and open set, the hyperplane separation result in Corollary 2.8 yields the existence of anα∈R^k\ {0}and a c∈Rwith

hy, αi_Rk ≥c≥ hx, αi_Rk

for all y∈Y +R^k≥−y¯and allx ∈R^k<. Because ofhx, αi

R^k ≤cfor all x∈R^k< it followsα ≥0.

But then we immediately obtainc= 0. Therefore, for all y∈Y and all x∈R^k≥ we get hy+x−y, αi¯

R^k ≥0, and by settingx= 0 this yields

hy, αi

R^k ≥ h¯y, αi

R^k

for ally∈Y and hence

k

X

i=1

αifi(u)≥

k

X

i=1

αifi(¯u) for allu∈U_ad.

Thus, ¯u is a solution of (WSP)α and due to the unique solvability of this optimization problem

¯

uhas to be the unique solution.

In the following we want to investigate how the solution of a weighted sum problem depends on the weightsα.

Lemma 3.21. Let Assumption 1 be satisfied and assume that the functions f1, . . . , f_k are ad-ditionally twice differentiable such that ∇²f_i is positive definite for all i ∈ {1, . . . , k} as well as that there existsip ∈ {1, . . . , k} such that ∇²fip is uniformly positive definite with coercivity constant Cip. If α ∈ R^k with αip >0, then ∇²Fα is uniformly positive definite with coercivity constant C_ip·α_ip.

Proof. Letu∈U_ad be arbitrary. Then we get

∇²F_α(u)v=

k

X

i=1

α_i∇²f_i(u)v for all v∈U,

and therefore

h∇²Fα(u)v, vi_U =

k

X

i=1

αih∇²fi(u)v, vi_U

≥C_ip·α_ipkvk²_U for all v∈U, which is the claim.

Assumption 3. Assume that the functionsf₁, . . . , f_k are twice continuously differentiable such that ∇²fi is positive definite for all i ∈ {1, . . . , k} as well as that there exists ip ∈ {1, . . . , k}

such that ∇²f_ip is uniformly positive definite with coercivity constant C_ip.

The following result assures us that the unique solution of the weighted sum problem depends continuously on the weights, if Assumptions 1 and 3 are satisfied.

Theorem 3.22. Let Assumptions 1 and 3 be satisfied and defineZ :={α ∈R^k |α_ip > κ} for an arbitrary κ >0. Then the mappingZ →Uad, α 7→u¯^α, where u¯^α is the unique minimizer of F_α, is locally Lipschitz continuous and thus in particular continuous.

Proof. Letα¹ = (α¹₁, . . . , α¹_k)^T, α² = (α²₁, . . . , α²_k)^T ∈ Z be arbitrary and denote by ¯u¹and ¯u² the unique minimizers ofF_α¹ and F_α², respectively. If we plugu= ¯u² into the first-order condition (3.2) for F_α¹ and u= ¯u¹ into the first-order condition (3.2) forF_α² and add both inequalities, we end up with

0≤ h∇F_α1(¯u¹),u¯²−u¯¹i_U+h∇F_α2(¯u²),u¯¹−u¯²i_U

=−h∇F_α1(¯u¹)− ∇F_α1(¯u²),u¯¹−u¯²i_U+h∇F_α2(¯u²)− ∇F_α1(¯u²),u¯¹−u¯²i_U. (3.4) Using the mean value theorem for the first term, we can conclude that there exists ˆu with

−h∇F_α1(¯u¹)− ∇F_α1(¯u²),u¯¹−u¯²i_U =−h∇²F_α¹(ˆu)(¯u¹−u¯²),u¯¹−u¯²i_U, where using the uniform positive definiteness shown in Lemma 3.21 yields

−h∇²F_α¹(ˆu)(¯u¹−u¯²),u¯¹−u¯²i_U ≤ −C_ip·κ

u¯¹−u¯²

2 U. Plugging this into (3.4) implies

Cip·κ

u¯¹−u¯²

2

U ≤ h∇F_α2(¯u²)− ∇F_α1(¯u²),u¯¹−u¯²i_U. (3.5) For the term on the right-hand side we get by using the Cauchy-Schwarz inequality twice

h∇F_α2(¯u²)− ∇F_α1(¯u²),u¯¹−u¯²i_U =

k

X

i=1

(α²_i −α¹_i)h∇f_i(¯u²),u¯¹−u¯²i_U

≤

¯u¹−u¯² _U

k

X

i=1

(α²_i −α_i¹)

∇f_i(¯u²) _U

≤

k

X

i=1

∇f_i(¯u²)

2 U

!¹₂

α¹−α² R^k

u¯¹−u¯² U.

Altogether we obtain

u¯¹−u¯²

U ≤C(α²)

α¹−α² R^k, whereC(α²) := (C_ip·κ)⁻¹

Pk i=1

∇f_i(¯u²)

2 U

¹₂

is independent of ¯u¹ and therefore ofα¹, which implies the local Lipschitz continuity of the mappingZ →U_ad,α7→u¯^α.

Geometrical Properties

The following insight will be needed later, when an algorithm is developed which approximates the Pareto front numerically. It is the application of Remark 3.9 to the weighted sum method.

Lemma 3.23. To α ∈ R^k let u¯ ∈ U_ad be the solution of (WSP)α and y¯ := f(¯u). Then the hyperplane H_α :={¯y+x| hx, αi

R^k = 0} is always below the objective feasible region Y (in the sense of Definition 2.2) with only boundary point y¯and hence also below the Pareto front PY. Proof. We need to show that

∀z∈H_α∀y∈Y : ¬(yz) (3.6)

∀y∈Y ∃z∈H_α : z≤y (3.7)

hold. Equation (3.6) is fulfilled, as ¯u is the solution of (WSP)α and Hα =g_α⁻¹({g_α(¯y)}), where g_α :R^k→R, g(x) :=P_k

i=1α_ix_i. This also shows that ¯y is the only boundary point.

To show (3.7), let y ∈ Y be arbitrary and define ˜y := y−y, i.e.¯ y = ¯y+ ˜y. Furthermore, let {α, ϕ₂, . . . , ϕk} be an orthonormal basis of R^k. Then it holds ˜y=hy, αi˜ _Rkα+Pk

i=2h˜y, ϕii_Rkϕi. Additionally, we get

h˜y, αi_Rk =hy−y, αi¯ _Rk =hy, αi_Rk − h¯y, αi_Rk ≥0, as ¯uis the minimizer ofF_α, and withx:=Pk

i=2h˜y, ϕ_ii

R^kϕ_iby definition of an orthonormal basis hx, αi_Rk = 0.

So in totaly = ¯y+x+h˜y, αi

R^kα with ¯y+x∈Hα and hy, αi˜

R^kα≥0. Thus, the claim follows.

Remark 3.24. As mentioned in Remark 3.9, the statement of Lemma 3.23 is the best geo-metrical information we can hope for. In this sense the weighted sum method yields optimal geometrical information.

By combining the ideas of Corollary 3.12 and Lemma 3.23, we obtain an analogue result for more general scalarization methods.

Lemma 3.25. Let Assumptions 1 and 2 be satisfied and let the function g : R^k_≥yid → R fulfil Presumption 1 and be additionally differentiable. Denote by u¯ the unique minimizer of F^g and define y¯:= f(¯u). If ∇g(¯y) 6= 0, then the hyperplaneH_g := {¯y+x | hx,∇g(¯y)i

R^k = 0} is below the objective feasible region and hence below the Pareto front with only boundary pointy.¯ Proof. First of all, by the strict monotonicity and the differentiability ofg, we can conclude that it holds ∇g(x) ≥0 for all x ∈ R^k_≥yid. This yields ∇g(¯y) 0. By defining α := ∇g(¯y), we can use Corollary 3.12 to conclude that ¯u also is the unique solution of (WSP)_α. Now, the claim follows from Lemma 3.23.

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