4th Non linear programming

• objective- constraint are non linear functions of the decision variables

• non-constant return to scale, effect from input on output is nonlinear

• in pricing models, price determines quantitaly sold (lower price more demand e.g.) -> nonlinear function

• to meaure goodness of the fit the sum the squared differences between the observed values nd the model’s prediction -> then they attempt to minimize this sum of squared differences -> quaring leads to nonlinearity

• financial models aim to achieve high return and low risk. Ris is usally measured in variance or std of portfolio -> nonlinear function of the decision variables (investment amount)

• non-lineasr more common, probably very difficult to find problems that are truly linear

• either approximate NLP with LP

• or by allowing nonlinearities in your models you can create more realistic models -> more difficult to solve

• with Solver this means the found solution may be suboptimal

• solver is guaranteed to solve certain types of NLPs correctly

  Convex: slope’s rate of change is always nondecreasing/ increasing

Concave: slope’s rate of change is always nonincreasing/ decreasing

Conditions for maximum problems

• objective function or it’s logarithm is concave

• constraints are linear

Conditions for minimization problem:

• the objective function is convex

• constraints are linear

• For the figure graphed below, points A and C are called local maxima because the function is larger at A and C than at nearby points.

• However, only point A maximizes the function; it is called the global maximum.

• The problem is that Solver can get stuck near point C, concluding that C maximizes the function.

1. Start at an initial point

   Solver chooses a starting solution (your initial guess).

2. Compute the gradient

   It estimates how the objective function changes in each direction and identifies the steepest descent direction (for minimization).

3. Take a small step (line search)

   Solver moves a little in the downhill direction and checks if the objective improves.

   • If improvement → continue in that direction

   • If not → adjust step size or direction

4. Adjust for constraints (reduced gradient)

   If constraints exist, Solver modifies the movement so it stays inside the feasible region, like sliding along walls instead of going through them.

5. Repeat

   This loop (compute → step → check → adjust) repeats many times until:

   • improvement becomes tiny

   • or the gradient is near zero

     → Solver concludes it has reached a local optimum.

   
   

1. Try several possible starting values for the changing cells,

2. Run Solver from each of these, and

3. Take the best solution Solver finds.

-> does not guarantee the optimal solution as well

• We turn the two-objective “maximize return & minimize risk” portfolio problem into a single-objective nonlinear problem by fixing a minimum return and then minimizing variance.

• Step 1 — Choose a minimum required expected return

  Example:

  “Portfolio must earn at least 10%.”

  This becomes a constraint

• Step 2 — Minimize the variance (risk)

  Now the objective is only one:

  Minimize Risk

• Specifically, if we believe that stock returns are approximately normally distributed, then the probability is about 0.68 that the actual portfolio return will be within one standard deviation of the expected return,

• and the probability is about 0.95 that the actual portfolio return will be within two standard deviations of the expected return.