Nonlinear Programming

1
Nonlinear Programming

• In this handout
• Situations where nonlinear programs can be
  applied
• Graphical illustration of nonlinear programs
• Types of nonlinear programs

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Recall types of Optimization Models
Stochastic (probabilistic information on data)
Deterministic (data are certain)
Discrete, Integer (S Zn)
Continuous (S Rn)
Linear (f and g are linear)
Nonlinear (f and g are nonlinear)
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Nonlinear programming

• General form
• Find x1,,xn so as to
• min or max f(x1,,xn) (objective function)
• subject to gi(x1,,xn) bi (functional
  constraints)
• x1,,xn ? S (set constraints)
• where at least some of the f and gi functions are
  nonlinear.
• There are different types of nonlinear programs,
  depending on the characteristics of the f and gi
  functions.

4

• Some situations when nonlinear programming can be
  applied.
• In product-mix problem, can have
• Price elasticity, whereby the amount of a product
  that can be sold has an inverse relationship to
  the price charged.
• Marginal cost of production varies with the
  production level. Marginal cost may decrease
  because of a learning-curve effect (more
  efficient production with more experience).
• In transportation problem, volume discounts are
  available for large shipments.

5
Graphical illustration of nonlinear programs
An example with nonlinear constraints when the
optimal solution is not a corner point feasible
solution.
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Graphical illustration of nonlinear programs
An example with linear constraints but nonlinear
objective function when the optimal solution is
not a corner point feasible solution.
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Graphical illustration of nonlinear programs
An example when the optimal solution is inside
the boundary of the feasible region.
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Graphical illustration of nonlinear programs
An example when a local maximum is not a global
maximum (the feasible region is not a convex set).
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Types of Nonlinear Programming problems

• Unconstrained optimization
• min or max f(x1,,xn)
• No functional constraints.
• Linearly constrained optimization
• Objective function nonlinear
• Functional constraints linear
• Extensions of simplex method can be applied.
• Quadratic programming
• Special case of linearly constrained
  optimization when the objective function is
  quadratic.

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Types of Nonlinear Programming problems

• Convex programming
• Objective function f is concave
• Each gi is convex
• Covers a broad class of problems.
• A local maximum is a global maximum.

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Types of Nonlinear Programming problems

• Separable programming
• A special case of convex programming when f and
  gi are separable functions. In a separable
  function each term involves just a single
  variable.
• E.g., f(x1, x2) x12 2x1- 4x22 3x2,
• Can be closely approximated by a linear
  programming problem.

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Types of Nonlinear Programming problems

• Nonconvex programming
• Even if we are successful in finding a local
  maximum, there is no assurance that it also will
  be a global maximum.
• In some special cases (Geometric programming,
  Fractional programming), the problem can be
  reduced to an equivalent convex programming
  problem.