Mathematical Background

1

Trail course Variational Inequalities

• Chapter 2
• Mathematical Background
• Karel Lindveld
• Delft University of Technology
• Faculty of Civil Engineering and Geo Sciences
• Transportation Planning and Traffic Engineering
  Section

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Contents of chapter 2 presentation outline

• The VIP and related problems
• Definitions
• Related problems
• Optimization problems
• (CP, SE)
• Fixed point problems
• Existence and uniqueness results
• Solution algorithms for VIPS
• The general iterative scheme
• Specific instances of the general iterative
  scheme
• The diagonalisation method
• The projection method, modified projection method
• The Frank-Wolfe method for constrained
  optimisation

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Remark

• Chen last sentence on page page 36 is an
  understatement

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VI (1) Are u and c matrices or vectors?
From a mathematical point of view u and c only
have vector properties
Chen matrix form for u (convenient notation)
But potentially misleading !
Easy to re-write u as a vector
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VI (2) Matrices with vector properties
Dependency of c on u
Multiplication not explicitly denoted
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VI (3) inner products
Apparently our VIs are formulated in terms of
inner products on
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VI (4) Inner products define cones
The set
Is the cone of vectors that have an obtuse angle
with c
x
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Vector subtraction
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Equivalent problems and special case
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NLP (1)
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NLP (2) equivalence
Minimisation problem
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Positive (semi) definiteness and convexity
Consider the second order Taylor series
approximation of f
Theorem H(x) is positive semi-definite for x in
S ? f is convex
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Convexity gives uniqueness
Then
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Positive semidefiniteness and ellipsoid forms
Connection with quadratic forms Let A be
symmetric (note a Hessian is always symmetric),
then A can be reduced to diagonal form by a
unitary coordinate transform (that is
rotations and mirrorings)
With respect to this coordinate system the
quadratic form
becomes
which is an ellipse
Therefore if the Hessian of F at a point is psd,
then F is locally ellipse-shaped
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Convexity and monotonicity
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Relationship between function levels
Symmetry check level
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Levels in user optimum and system optimum
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Solution methods for VIPs

• General iterative algorithm
• cant solve VIP directly, but can solve NLPs
• How can this help ???
• Assume cost has asymmetric Jacobian, so cant be
  recast as NLP
• Answer construct sequence of auxiliary VIPs
• each VIP has symmetric Jacobian, and corresponds
  to NLP
• solution of previous VIP is a parameter for next
  one
• make sure that sequence of solutions to auxiliary
  VIPs (I.e. NLPs) solves the original VIP
• How to construct this sequence?
• See Chen

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The diagonalisation method
Apply F with one argument fixed, one variable
Jacobian symmetric, can therefore formulate VIP
as NLP problem
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The projected gradient method
See Chen
Note that
Check that
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Solution methods for NLP problems
(see Bazaraa)
Constrained minimisation problems NLPs
Unconstrained minimisation problems
Constrained minimisation problems NLPs with
linear constraints
Penalty and barrier functions Methods of
feasible directions . gradient
projection
Newtons method BFGS Trust-region methods
Frank-Wolfe
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Solution methods for VIPs
Original VIP problem
Sequence of auxiliary NLP problems
Sequence of Linear programming problems (Shortest-
path problem)
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Discussion, questions