Optimization unconstrained and constrained

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Optimizationunconstrained and constrained

• Calculus part II

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Setting-Up Optimization Problems

• Define the agents goal objective function and
  identify the agents choice (control) variables
• Identify restrictions (if any) on the agents
  choices (constraints). If no constraints exist,
  then we have unconstrained minimization or
  maximization problems.
• If constraints exist, what type?
• Equality Constraints (Lagrangian)
• Inequality Constraints (Linear Programming)

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• Mathematically, Optimize y f(x1, x2, . . .
  ,xn)
• subject to (s.t.)
• gj (x1, x2, . . . ,xn) bj
• or
• bj j 1, 2, . . ., m.
• or
• bj
• y f(x1, x2, . . . ,xn) ? objective function
• x1, x2, . . . ,xn ? set of decision variables
  (n)
• optimize ? either maximize or minimize
• gi(x1, x2, . . . ,xn) ? constraints (m)

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• Constraints refer to
• restrictions on resources
• legal constraints
• environmental constraints
• behavioral constraints

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Review of Derivatives

• yf(x) First-order condition
• Second-order condition
• Constant function
• Power function
• Sum of functions
• Product rule
• Quotient rule
• Chain rule

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Unconstrainted UnivariateMaximization Problems
max f(x)

• Solution
• Derive First Order Condition (FOC) f(x)0
• Check Second Order Condition (SOC) f(x)lt0
• Local vs. global If more than one point satisfy
  both FOC and SOC, evaluate the objective function
  at each point to identify the maximum.

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Example

• PROFIT -40 140Q 10Q2
• Find Q that maximizes profit

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Example

• PROFIT -40 140Q 10Q2
• Find Q that maximizes profit
• 140 20Q set 0
• Q 7
• - 20 lt 0
• max profit occurs at Q 7
• max profit -40 140(7) 10(7)2
• max profit 450

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Minimization Problems Min f(x)

• Solution
• Derive First Order Condition (FOC) f(x)0
• Check Second Order Condition (SOC) f(x)gt0
• Local vs. global If more than one point satisfy
  both FOC and SOC, evaluate the objective function
  at each point to identify the minimum.

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Example

• COST 15 - .04Q .00008Q2
• Find Q that minimizes cost

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Example

• COST 15 - .04Q .00008Q2
• Find Q that minimizes cost
• -.04 .00016Q set 0
• Q 250
• .00016 gt 0
• Minimize cost at Q 250
• min cost 10

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Unconstrained Multivariate Optimization

• Max
• FOC
• SOC

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Example
Find Q1 and Q2 that maximize Profit
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Example

• PROFIT is a function of the output of two
  products
• (e.g.heating oil and gasoline)
• Q1 Q2
• Solve Simultaneously Q1 5.77 units
• Q2 4.08 units

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Second-Order Conditions
(-20)(-16) (-6)2 gt 0 320 36 gt 0 we
have maximized profit.
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Constrained Optimization

• Solution Lagrangian Multiplier Method
• Maximize y f(x1, x2, x3, , xn)
• s.t. g(x1, x2, x3, , xn) b
• Solution
• Set up Lagrangian
• FOC

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Lagrangian Multiplier

• Interpretation of Lagrangian Multiplier ? the
  shadow value of the constrained resource.
• If the constrained resource increases by 1 unit,
  the objective function will change by ? units.

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Example

• Maximize Profit
• subject to (s.t.) 20Q1 40Q2 200
• Could solve by direct substitution
• Note that 20Q1 200 40Q2 ? Q1 10 2Q2
• Maximize Profit

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Lagrangian Multiplier Method
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