BASIC CONCEPTS IN OPTIMIZATION: PART III: Continuous

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BASIC CONCEPTS IN OPTIMIZATION PART III
Continuous Constrained
Important concepts for the optimization of
systems with continuous variables and non-linear
equations.

• Convert to Unconstrained - Penalty functions
• Concept of Equality Constrained Optimization
• - Linear equalities / Non-linear equalities /
  Lagrangian
• Concept of Inequality Constrained Optimization
• - Definition of optimum / Lagrangian
• Optimality Conditions - KKT

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CONSTRAINED OPTIMIZATION
CLASS EXERCISE Is this system constrained or
unconstrained?
- The goal is to maximize CB in the effluent at
S-S - You can adjust only the flow rate of feed

• This is an isothermal CFSTR with the reaction
• A ? B ? C
• You can adjust F, CA0
• You can adjust F, CA0, Fc, and V

Before we solve it, can we recognize it?
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CONSTRAINED OPTIMIZATION
CLASS EXERCISE Is this system constrained or
unconstrained?
Optimization of XD and XB - energy/yield
tradeoff can be unconstrained in some situations
Increasing energy
XB, impurity
0
0
XD, impurity
Why might this be constrained?
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CONSTRAINED OPTIMIZATION
CLASS EXERCISE Is this system constrained or
unconstrained?
Optimization of XD and XB - energy/yield
tradeoff can be unconstrained in some situations
Reboiled vapor
pressure
Please explain the constraints. We need to model!
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CONSTRAINED OPTIMIZATION
CLASS EXERCISE Is this system constrained or
unconstrained?
Optimization of XD and XB - energy/yield
tradeoff can be unconstrained in some situations
Reboiled vapor
pressure
Now, where should we operate?
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
HARD SOFT CONSTRAINTS Convert inequality
constraints to terms in the objective function
that force the solution into (at least towards)
the feasible region.
The penalty parameter, r, can be adjusted using
an iterative method.
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
feasible
External Penalty Functions change the objective
in the infeasible region. For example,
F(x)
f(x)
Increasing r
x
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Constrained Optimization Penalty Approaches
Convert to Unconstrained

• The power of violation lt 1, likely too weak a
  penalty
• The power of violation 1, can match
  unconstrained optimum, but discontinuous
  derivatives
• The power of violations gt 1 note that 2 gives
  continuous derivatives

External Penalty Functions
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
Increasing r
Internal Penalty Functions change the objective
in the feasible region. For example,
F(x)
f(x)
feasible
x
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
Internal Penalty Functions

• Requires a feasible starting point and cannot
  have infeasible point at any iteration
• Constraints should be normalized to equally
  penalize
• Must modify r during iterations

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Constrained Optimization Penalty Approaches
Convert to Unconstrained
CLASS EXERCISE Convert the following constrained
optimization to an unconstrained optimization.
A ? B ? C
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
CLASS EXERCISE This solution applies the
external penalty function, r.
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Constrained Optimization Penalty Approaches
Convert to Unconstrained

• The penalty can strongly distort the contours, so
  it is not typically used to convert the
  constrained to unconstrained problem.
• However, we will use these concepts in some
  algorithms.

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Constrained Optimization Penalty Approaches
Convert to Unconstrained
From Reklaitis et al, 1983
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
From Reklaitis et al, 1983
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
From Reklaitis et al, 1983
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
From Reklaitis et al, 1983
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
INEQUALITY 5-X1-X2gt0
From Reklaitis et al, 1983
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Constrained Optimization Penalty Approaches
Convert to Unconstrained
From Reklaitis et al, 1983
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Constrained Optimization Directional Derivative
Convert to Unconstrained
GENERAL CONCEPT FOR EQUALITY CONSTRAINED
OPTIMIZATION The constraints introduce
limitations on the allowable moves in the
variables (?x). For equality constrained
problems, the moves must remain on the curve of
the constraint.
Equality constraint

• How many DOF?
• How do we determine the gradient of profit?

x2
Can we move in this direction?
x1
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Constrained Optimization Directional Derivative
Convert to Unconstrained
x is a vector of variables (flows, compositions,
etc. ) It has a dimension n.
Dimension n-m
Dimension m

• If we have n variables and m (independent)
  EQUALITY constraints, we have n-m degrees of
  freedom for optimization.
• We can solve the equations analytically to
  eliminate m of the variables.

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Constrained Optimization Directional Derivative
Convert to Unconstrained
y is a vector of variables (flows, compositions,
etc. It has a dimension n-m. After solving for
y, we can solve for z Z(y)

• We must be able to analytically solve for z as a
  function of y and substitute these into the
  original problem.
• This is always possible for linear equations, not
  so for non-linear equations.

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Constrained Optimization Directional Derivative
Convert to Unconstrained
CLASS EXERCISE Convert the following constrained
optimization to an unconstrained optimization and
solve.
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Constrained Optimization Directional Derivative
Convert to Unconstrained
CLASS EXERCISE Start by eliminating one variable
using the linear equation.
minimum
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Constrained Optimization Directional Derivative
Convert to Unconstrained
What if the equations are not linear The
constraints introduce limitations on the
allowable moves in the variables (?x). For
equality constrained problems, the moves must
remain on the curve of the constraint. Or, at
least near the constraint
Equality constraint
x2
Can we move in this direction?
x1
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Constrained Optimization Directional Derivative
Convert to Unconstrained
The total derivative of the objective function is
why?
The total derivative of the constraint must be
zero.
How can we use these results to determine the
constrained derivative?
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Constrained Optimization Directional Derivative
Convert to Unconstrained
Rearrange the constraint equation to solve for
dx1 (we could have solved for dx2).
The derivatives are evaluated at a point (x1,
x2) dh/dx2 ? 0
This is the relationship between x1 and x2 that
is forced by the equality constraint at a
specific point.
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Constrained Optimization Directional Derivative
Convert to Unconstrained
Replace with result from constrained change
Directional derivative or reduced gradient is
derivative while observing the equality
constraint(s) at a point
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Constrained Optimization Directional Derivative
Convert to Unconstrained
All points on this curve satisfy h0
f
x1
NECESSARY CONDITION FOR OPTIMALITY From the basic
concept of optimality, the directional or reduced
gradient must be zero for a minimum. (This is
not sufficient).
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Constrained Optimization Directional Derivative
Convert to Unconstrained
CLASS EXERCISE Express the necessary condition
for the equality constrained minimum of the
following problem.
at x 3 4
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Constrained Optimization Directional Derivative
Convert to Unconstrained
CLASS EXERCISE The method finds a search
direction that approximately remains on the
equality constraint.
How far should we go in this direction?
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Constrained Optimization Directional Derivative
Convert to Unconstrained
CLASS EXERCISE The method finds a search
direction that approximately remains on the
equality constraint.
How far should we go in this direction?
x2
f(x)
h0
x1
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Constrained Optimization Directional Derivative
Convert to Unconstrained
GENERALIZE THE CONDITION FOR A MINIMUM FOR MANY
EQUALITY CONSTRAINTS AND MANY VARIABLES.
These are the same conditions we have used
previously, but they are in a reduced space of
moves in x that satisfy h(x)0.
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Constrained Optimization Lagrangian Convert to
Unconstrained
Lets reconsider the equality constrained problem.
We can define the term to be lambda
Necessary conditions
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Constrained Optimization Lagrangian Convert to
Unconstrained
The stationarity for the equality constrained
problem
Identical!
Can be restated as the stationarity of the
Lagrangian
Identical!
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Constrained Optimization Lagrangian Convert to
Unconstrained
The stationarity for the equality constrained
problem occurs at the same values of x as the
stationarity of the Lagrangian!
Definition
Stationarity
? is the Lagrange multiplier its value is
determined by the stationarity conditions.
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Constrained Optimization Lagrangian Convert to
Unconstrained
CLASS EXERCISE Express the necessary condition
for the equality constrained minimum of the
following problem using Lagrange multipliers.
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Constrained Optimization Lagrangian Convert to
Unconstrained
CLASS EXERCISE Determine the stationarity
equations.
h 0
f

These non-linear equations define the
stationarity points x1 2.77 -2.77 x2
4.16 -4.16 ? 0.36 -0.36
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Constrained Optimization Lagrangian Convert to
Unconstrained
CLASS EXERCISE The method finds a search
direction that approximately remains on the
equality constraint.
Which of the stationary points is a minimum?
x2
f(x)
h0
x1
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Constrained Optimization Lagrangian Convert to
Unconstrained
How can we interpret the Lagrange multiplier?
Original problem with rhs isolated
Lagrangian
Stationarity or necessary conditions
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Constrained Optimization Lagrangian Convert to
Unconstrained
Lets simplify to two x variables and one
equality
A
B
1
Multiply B by ? and subtract from A.
What is the result if we evaluate this at the
stationary point?
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Constrained Optimization Lagrangian Convert to
Unconstrained
At the optimum
The Lagrange multiplier is the sensitivity of the
objective to the rhs - at the optimum!
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Constrained Optimization Lagrangian Convert to
Unconstrained
We have just covered an important and complex
issue.
Lets look back and review the material in a less
formal manner, to see the essential logic and
simplicity, and to develop a graphical
interpretation.
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It is easy, I like it!!
It is easy, I like it!!
Constrained Optimization Comparing directional
derivatives and Lagrangian
Lets look at the linearly constrained problem
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Lets look at the non-linearly constrained problem
This is a NL problem, with non-linear constraints
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Non-linear equality constraint, h(x) 0
x2
This is the direction of steepest descent, ?x f
x1
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Lets look ahead an see how we will use this
funny derivative.
Next, we perform a linear search along the
direction and find a local, approximate
minimum Note we use the non-linear equation at
each step along the direction.
x2
Non-linear equality constraint, h(x) 0
x1
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Next, we can return to the constraint we must
solve some NL equations.
x2
x
x
x
x1
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Non-linear equality constraint, h(x) 0
x2
Then, we evaluate the directional derivative
again and repeat the process
x1
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Oh, no!
Constrained Optimization Comparing directional
derivatives and Lagrangian
Lets look at the Lagrangian formulation
This is a NL problem, with non-linear constraints
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Lets look at the Lagrangian formulation
necessary conditions for optimality (stationarity
conditions).
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That wasnt so bad!!
Constrained Optimization Comparing directional
derivatives and Lagrangian
Lets look at the Lagrangian formulation
It can be applied to linear or non-linear
constraints
It can be extended to inequality constraints
There is no sign restriction on the lagrange
multiplier of an equality constraint
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Visual display of constrained optimization
f Unconstrained objective function
f
x2
Steepest descent for unconstrained optimization
x1
h(x)0
Ignored for the unconstrained problem
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Constrained Optimization Comparing directional
derivatives and Lagrangian
Visual display of constrained optimization
L The Lagrangian, which is stationary at the
optimum
L
x2
Steepest descent for constrained optimization
x1
h(x)0

• Caution
• This a 2-dimensional picture of a 3-dimensional
  system
• The correct lagrange multiplier values are only
  known at the optimum other values are estimates.

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Constrained Optimization Comparing directional
derivatives and Lagrangian
Visual display of constrained optimization
L
x2
All points on this curve satisfy h(x1, x2)0
f2(x1)
f2(x1)
x1
h(x)0
x1
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Constrained Optimization Inequality Constrained
Basic Concepts
GENERAL CONCEPT FOR INEQUALITY CONSTRAINED
OPTIMIZATION
feasible
feasible
When the constraints are not active, no change.
What is the condition when a constraint is active?
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Constrained Optimization Inequality Constrained
Basic Concepts
QUICK REVIEW FOR UNCONSTRAINED Basic Definition
The general definition of a minimum of f(x) is x
is a minimum if f(x) ? f(x ?x) for small ??x
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Constrained Optimization Inequality Constrained
Basic Concepts
FOR INEQUALITY CONSTRAINED Basic Definition For
a minimum, all feasible points around the minimum
have objective values higher than at the minimum.
x2
feasible
x1
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Constrained Optimization Inequality Constrained
Basic Concepts
FOR INEQUALITY CONSTRAINED
What goes here? , ? , ?
What limitations are placed on the ?x vector?
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Constrained Optimization Inequality Constrained
Basic Concepts
FOR INEQUALITY CONSTRAINED
For minimizing
Restrict the ?x vector to feasible directions.
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Constrained Optimization Inequality Constrained
Basic Concepts
CLASS EXERCISE Graphically represent the
conditions for the optimum of the system sketched
below.
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Inequality Constrained Optimization Equality and
Inequality Constrained Optimality conditions
We want to extend the Lagrangian for inequality
constraints to formulate optimality
conditions. 1. Define a Lagrangian that
combines the objective and constraints - both
equality and inequality 2. Equality constraints
are always active 3. For each inequality
constraint, we need to account for two situations
- active or inactive 4. Find stationarity
conditions for this unconstrained
Lagrangian The approach must turn the
constraint on (active) or off (inactive) for the
correct stationarity condition
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Inequality Constrained Optimization Equality and
Inequality Constrained Optimality conditions
Objective function
Equality constraints
Inequality constraints
x Problem variables (vector) ? Lagrange
multipliers for equalities (vector) u Lagrange
multipliers for inequalities (vector)
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Inequality Constrained Optimization Equality and
Inequality Constrained Optimality conditions
? Lagrange multipliers for equalities
Shadow price for the constraint u Lagrange
multipliers for inequalities Shadow price
for the active constraints
Complementarity conditions
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Inequality Constrained Optimization Equality and
Inequality Constrained Optimality conditions
Discuss the interpretation of Lagrange multiplie
rs

• These are the complementarity conditions.
• When gk(x)gt 0 inactive, its Lagrange multiplier
  uk 0
• When gk(x) 0 active, its Lagrange
  multiplier uk gt 0

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Inequality Constrained Optimization Equality and
Inequality Constrained Optimality conditions

• We note the following important properties
• We have transformed a constrained to an
  unconstrained problem with variable bounds
• L(x, ?, u) f(x) for feasible x
• A local minimum of L(x, ?, u) occurs at the
  local minimum of f(x) thus, we determine the
  x and the sensitivities of the constraints, ?,
  u

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General Constrained Optimization
Karesh-Kuhn-Trucker Conditions
Necessary sufficient conditions for optimality
(ga(x)active)
Stationarity curvature
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General Constrained Optimization
Karesh-Kuhn-Trucker Conditions
CLASS EXERCISE What is the meaning of the
requirement that the Lagrange multipliers are
positive?
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General Constrained Optimization
Karesh-Kuhn-Trucker Conditions
CLASS EXERCISE Determine the necessary
(stationarity) conditions for a minimum in the
following problem.
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General Constrained Optimization
Karesh-Kuhn-Trucker Conditions
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General Constrained Optimization
Karesh-Kuhn-Trucker Conditions

• Solutions evaluated in the table.
• Are these stationary points?
• What is the nature of each?
• Interpret the Lagrange multipliers

Small roundoff error
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General Constrained Optimization
Karesh-Kuhn-Trucker Conditions

• For the famous Karesh-Kuhn-Trucker (KKT)
  Conditions for optimality.
• The functions must be twice continuously
  differentiable
• The active constraints must be linearly
  independent at the optimum
• The result defines a local minimum

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Constrained Optimization Basic Concepts to an
Algorithm

• Using the Lagrangian to design NLP solvers.
• We will apply the concepts of the unconstrained
  optimizers to the Lagrangian.
• However, we must be careful about measuring
  progress, because L k1 lt Lk ensures that f k1
  lt fk only if the points are feasible.
• Therefore, our methods must ensure feasibility or
  devise a measure of improvement (merit) that
  distorts the geometry to create min(L) min f
  at the same x.

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BASIC CONCEPTS III - Workshop 1
The notes include a workshop evaluating the
Lagrangian for the following problem.
Plot the response surface, f(x), and the
constraint. Based on your plot, confirm the
results from the class workshop and determine the
point where f(x) is maximum.
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BASIC CONCEPTS III - Workshop 2
The figure shows a minimization problem with an
equality constraint. Sketch the feasible
directions reducing f(x) for several points on
the equality constraint. Based on your sketch,
show that the following equation is true.
Hint The feasible direction is perpendicular to
?xh(x).
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BASIC CONCEPTS III - Workshop 3
The KKT conditions require the stationarity of
the Lagrangian at an optimum point, x.
Is the following condition required at a minimum?
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BASIC CONCEPTS III - Workshop 4
The KKT conditions provide necessary conditions
for an optimum point, x.
A non-linear optimization method could solve the
resulting equations. Discuss this approach for
optimization do you recommend it?
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BASIC CONCEPTS III - Workshop 5
Lets think about a new method for optimizing
constrained, linear problems, an LP. The famous
Simplex method finds improvements in adjacent
corner points, following the edges.
Alternatively, we could apply the internal
penalty function and find improving directions in
the interior of the region. Discuss this
alternative. - Is it possible? - Might any
advantages exist?
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BASIC CONCEPTS III - Workshop 6

• We recognize the advantage of formulating a
  convex programming problem, i.e., a local optimum
  is a global optimum.
• Is a general non-linear objective function
  convex?
• Under what conditions is a feasible region
  defined by
• inequality constraints convex?
• Hint Apply the definition of convexity to the
  objective function and to inequalities defining
  the feasible region.