Semidefinite Programming Application: Truss Topology Design

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Semidefinite Programming Application Truss
Topology Design

• Jean-Christophe Lilot
• Ting Ting Ren
• CAS735, Friday 2nd 9th April 2004

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Outline

• Basic TTD problem (Assignment 2)
• Limitation of the basic TTD
• Standard Static TTD model
• SemiDefinite Reformulation ? Primal P
• Robust model leads to Universal model ? Pr
• Simplification Dl of the Dual D of Pr
• Comparison of Pr, dual of Dl, and Pr

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Basic TTD model Presentation

• Given
• A set of tentative bars
• A set of fixed nodes
• An upper bound of the quantity of material
• A load F to apply to the structure.
• The property of the material
• Design the most stable truss topology design

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Basic TTD model Formulation
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Limitations of the model

• No geographical limitations
• Not enough material limitations
• Can this model be called robust?

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Geographical limitation
3
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Material limitation

• What we have
• Upper and lower bounds on the total amount of
  material
• What we need
• Upper bound on the bar diameter
• Lower bound on the bar diameter

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Robustness Problem
Compliance 1.000
Compliance 8.4
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Robustness Solutions

• Take into account several load scenarios

• Take into account all small occasional loads that
  may apply to the system

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Robustness New problems

• Mathematical point of view

• Engineering point of view

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Robustness New solutions

• The set of force F can be the ellipsoidal
  envelope of the original finite set of loads and
  a small ball
• We use a two stage scheme.
• First we dont include the ellipsoidal envelope
  to get the important nodes
• Then we treat the set of nodes actually used by
  the preliminary truss as our new nodal set

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Robustness Results

• Euclidean norm of the ball not exceeding 10 of
  the norm of the original force.

• Compliance always less or equal to 1.03 within
  the elipsoid.
• Compliance with respect to the original load
  1.0024 so only 0.24 larger than the optimal
  compliance

Old Design
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Standard Static TTD Model S.1.

• The set kinematically admissible displacement is
  a polyhedral set
• The system of linear inequalities satisfies the
  Slater condition

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Standard Static TTD Model S.2.

• The set T of admissible designs is given by a
  single linear constraint on the vector t, namely
  upper and lower bounds on ti and an upper bound
  on the total material ressourcewith given
  parameters

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Standard Static TTD Model S.3.

• The set F of loading scenarios is either a finite
  set (multiload structural design)
• or an ellipsoid (robust structural design)

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Standard Static TTD Model S.4.

• Exclusion of rigid body motions of the ground
  structure

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Standard Static TTD Model The variation
principal
Proposition variationnal description of
compliance
Consider a ground structure (n, m, B1,, Bn)
along with a load , and let
be a truss. Let us associate with these data the
quadratic form (The potential energy of the
closed system) of . The compliance
is finite if and only if the form
below bounded on , and whenever it is the
case, one has
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Standard Static TTD Model The simple lemma
Lemma
Let be a quadratic
form on with symmetric positive
semidefinite matrix A. Then (i) The form
below is bounded if and only if it attains its
minimum (ii) The form attains its minimum
if and only if the equation is solvable, and if
it is the case, the set of minimizers of the
form is exactly the set of solutions to the
equation. (iii) The minimum value of the
form, if exists, is equal to being (any)
solution to .
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Standard Static TTD Model The simple Lemma ?
proof

• (i) There are two possibilities(a) b is
  orthogonal to Ker A if it is below bounded then
  it attains its minimum(b) b has a nonzero
  projection b onto Ker A below unbounded
• (ii) Since the form is convex and smooth, its
  minimizers are exactly the same as its critical
  points
• (iii)

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Errata

• Outline Not Pl, (Pr)
• 5th slide Its not the primal (Pr) but the dual
  (Dl)

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Standard Static TTD Model

• With
• A ground structure, i.e.,
• The space of virtual displacement along with
  its closed convex subset of kinematically
  admissible displacements,
• A collection .
• A set of admissible
  designs
• A set of loading scenarios,
• Find which minimize the worst compliance

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SDO reformulation I
We need a SDR of the compliance
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SDO reformulation II
Proposition
is satisfied if and only if there exists a
nonnegative vector µ of the dimension q equal to
the number of linear inequalities defining the set
of virtual displacements such that the matrix
is positive semidefinite. Thus the epigraph of
(regarded as a function of
and ) admits the SDR
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SDO reformulation IIIProposition proof
if
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SDO reformulation IIIProposition proof

• Lagrange duality
• Under the slater condition from S.1.
• for properly chosen is exactly
• the supremum of

Which is non negative
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SDO reformulation IIIThe simple Lemma
Lema
A quadratic inequality with a (symmetric) nxn
matrix A
is trivially true is valid for all -
if and only if the matrix
is positive semidefinite.
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SDO reformulation of the multiload TTD problem
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Second Model The ellipsoid
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Robust Model (Pr) Universal model!
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Link between the models

• Multiload model
• Robust model

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Standard Static TTD Model S.5.

• For every l1,,K, there existsand
  such that
• Pr is strictly feasible

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The primal (Pr)

• Dimension of (Pr)
• Multiload design of M-node structure
• Design dimention is O(M2)
• K big LMIs with row size of 2M or 3M

• Example 99 Nodes (1515 nodes)
• M81 (M225)
• 3200 vars (25100 vars)
• k LMIs of row size 160 or 240 (450 or 675)

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From Primal (Pr) to Dual (Dini) (step 0)
With design variables
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SimplificationFrom (Dini) to (D) (step 1)
With design variables
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SimplificationFrom (Dini) to (D) (step 1)
Eliminating and , we get
With design variables
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SimplificationFrom (D) to (D) (step 2)

• (D) is feasible
• We can add the following constraint without
  changing the opt valueending up with a new dual
  (D).
• Now, note that ifis a feasible sol. to (D),
  then the collectionis also a feasible sol. with
  the same objective
• From the LMI (D.a) by the Schur complement Lema
  it follows that , so that
  replacing by we preserve validity for the
  LMIs (D.f) and (D.a)

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SimplificationFrom (D) to (D) (step 2)
With design variables
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SimplificationFrom (D) to (D) (step 2)
Eliminating , we get
With design variables
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SimplificationFrom (D) to (Dl) (step n)
With design variables
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The Dual (Dl)

• Both Primal and Dual are strictly feasible ? both
  (Pr) and (Dl) are solvable with equal optimal
  values and with bounded level set.
• The optimal values are negation of each other.

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Case of a simple boundFrom (Dl) to (Dlsb)
With design variables
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Case of a simple boundFrom (Dl) to (Dlsb)
With design variables
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What we got!!!
M number of nodesk number of load
scenariosm 2M in plan and 3M in space
(Pr) in O(M6) (Dlsb) in O(k3M3)
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Back to the Primal (Pr)
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Equivalence between (Pr) and (Pr)
Proposition
A collection is a feasible
solution to (Pr) if and only if it can be
extended by properly chosen to a feasible
solution to (Pr)
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From (Pr) to (Pr)

• If part

Let a collection be a feasible sol to (Pr)
We want to prove the validity of the LMI (Pr.a)
For . We want
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From (Pr) to (Pr)
Using (Pr.d)
we get
Which is the value of the quadratic form with the
matrix (Pr.a) at the vector comprising x and
, and therefore is non negative.
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(Pr)
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From (Pr) to (Pr)

• Only if part

Let a be a feasible sol to
(Pr). Let us fix and set
For every the quadratic form of
is nonnegative, i.e., the equation
is solvable for every x.
We can chose its solution to be linear in x Then
i.e.
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From (Pr) to (Pr)
Let set then
With
Which proves (Pr.d)
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From (Pr) to (Pr)
We need to verify that for every collection
of vectors of appropriate dimension and for
every we have
We will prove the min of in
is non negative.
Given x, let set
minimize the partial derivative
with respect to is
We need to prove that
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From (Pr) to (Pr)
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A More Robust Problem?

• Back to the antenna design

A little bit more RobustSecond Order Conic
Optimization
?
Basic ProblemLinear Programming
Even more RobustSemidefinite Optimization
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A More Robust Problem?

• What about the TTD?

A little bit more RobustSemidefinite
Optimization
?
Basic ProblemSecond Order Conic Optimization
Even more RobustMust not be easily solvable!
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Thank you for your attention

• Have a good week-end!

References Course TextBookBenTal and A.
Nemirovskii Convex Optimization in Engineering,
SIAM Publications, 2001 Course slides
http//www.cas.mcmaster.ca/coe774/