LP Models • Know how to recognize an LP in verbose or matrix form; standard or otherwise; max or min • Know how to set up an LP • Understand how to make various conversions between different types of LPs (max/min; <, >, =; >, free; |abs. value|; etc.)
LP Modeling • Modeling Process • Understand the problem • Collect information and data • Identify and define the decision variables • formulate the objective function • Isolate and formulate the constraints
World Cup Portfolio Management • Portfolio Management for World Cup Assets • Five securities in a market for open trading at fixed prices and pay-offs, share limit on investment and short is not allowed • We’d like to decide how many shares to purchase to maximize the pay-off when the game is realized.
The Geometry of LPs • Understand the geometrical interpretation of LPs and associated intuition • Plot a feasible region in 2D • Plot the optimal iso-profit lines • Relate how the Simplex Method moves from corner point to adjacent corner point in improving direction • Understand the geographical interpretation of the various LP terms – (e.g. active constraints, basic variables, multiple optima, infeasibility, unbounded-ness,, etc.)
Theory of Linear Programming An LP problem falls in one of three cases: • Problem is infeasible: Feasible region is empty. • Problem is unbounded: Feasible region is unbounded towards the optimizing direction. • Problem is feasible and bounded: then there exists an optimal point; an optimal point is on the boundary of the feasible region; and there is always at least one optimal corner point (if the feasible region has a corner point). When the problem is feasible and bounded, • There may be a unique optimal point or multiple optima (alternative optima). • If a corner point is not “worse” than all its neighbor corners, then it is optimal.
Algebraic Interpretation of LPs • Understand why we emphasize basic solutions and basic feasible solutions (corner points). • Understand the equivalent (canonical) form of the LP and where it came from • Understand what the different elements of this form mean: reduced cost coefficient, shadow prices, reduced constraint matrix, RHS • Know when to terminate • Know its relation with dual
The Canonical Form of the LP • Recall the canonical form of the LP: all reduced cost coefficients for basic variables are zero, and reduced constraint matrix for basic variable form an identity matrix • Why do we represent the problem this way? • By looking at the objective function in terms of the non-basic variables only we can get insight into whether or not changing any of them might improve the objective function • By looking at the constraints in this way, we can see • on the RHS what values of the basic variables yield the feasible solution, and • in the constraint coefficients, how changing any non-basic variables will necessitate that we must change basic variables in response
Initial Tableau: x slack RHS Basis - c 0 0 A I b (>0) • Intermediate Tableau: Basis - c +y*A y* cBB-1b B-1A B-1 B-1b (0) Where y*= cBB-1, the dual solution to the production problem, B is the current basis selected from [ A I ] The Production Problem in Canonical Form
Sensitivity Analysis • Recall AGAIN the Simplex tableau and the canonical form and what it all means relative to sensitivity • Changing the cost vector and RHS • Know how far each element can be changed before we change to optimal basis • How do we find the effects on the optimal value and optimal solution while the basis remains optimal? • Don’t just memorize formulas! • Understand where they come from and why they work • Understand the importance of reduced costs and shadow prices for sensitivity
Reduced Cost and Objective Coefficient Range • All basic variables have zero reduced cost • In general, the reduced cost of any non-basic variable is the amount the objective coefficient of that variable would have to change, with all other data held fixed, in order for it to have a positive value in the optimum. • The objective coefficient ranges give the ranges of the objective function over which no change in the OS will occur. • One of the “allowable increase and decrease” for a non-basic variable is infinite and the other is the reduced cost. If a non-basic variable has zero reduced cost, then there exist alternative optimums.
Dual (Shadow) Price and Constraint RHS Ranges • All inactive constraints have zero dual price • In general, the dual price on a given active constraint is the rate of increase in the optimal value (OV) as the RHS of the constraint increases with all other data held fixed. If the RHS is decreased, it is the rate at which the OV is decreased. • The constraint RHS ranges give the ranges of the constraint RHS over which no change in the dual price will occur. • One of the “allowable increase and decrease” for an inactive constraint is infinite and the other equals the slack or surplus. • In general, when the RHS of an active constraint changes, both the OV and OS will change.
Duality • Know how to construct the dual • Understand the economic interpretation of y (=cBB-1) as shadow prices • Using this y, understand why the dual feasibility condition is the primal optimality condition • Know the duality theorems • Know the primal-dual optimality conditions and complementary slackness
General Rules for Constructing Dual 1. The number of variables in the dual problem is equal to the number of constraints in the original (primal) problem. The number of constraints in the dual problem is equal to the number of variables in the original problem. 2. Coefficient of the objective function in the dual problem come from the right-hand side of the original problem. 3. If the original problem is amax model, the dual is aminmodel; if the original problem is amin model, the dual problem is themaxproblem. 4.The coefficient of the first constraint function for the dual problem are the coefficients of the first variable in the constraints for the original problem, and the similarly for other constraints. 5. The right-hand sides of the dual constraints come from the objective function coefficients in the original problem.
General Rules for Constructing Dual ( Continued) 6. The sense of the ith constraint in the dual is = if and only if the ith variable in the original problem is unrestricted in sign. 7. If the original problem is man (min ) model, then after applying Rule 6, assign to the remaining constraints in the dual a sense the same as (opposite to ) the corresponding variables in the original problem. 8. The ith variable in the dual is unrestricted in sigh if and only if the ith constraint in the original problem is an equality. 9. If the original problem is max (min) model, then after applying Rule 8, assign to the remaining variables in the dual a sense opposite to (the same as) the corresponding constraints in the original problem. Max model Min modelxj 0 jth constraint xj≤ 0 jth constraint ≤ xj free jth constraint = ith const ≤ yi 0 ith const yi ≤ 0 ith const = yi free
Dual Pair Standard Primal Production Form: Dual Form (rotate primal 90o clockwise):
Primal-Dual in Matrix Form Standard Primal Production Form: Dual Form:
Primal-Dual in Matrix Form: Equality Standard (Equality) Primal Form: Dual Form:
Relations Between Primal and Dual 1. The dual of the dual problem is again the primal problem. 2. Either of the two problems has an optimal solution if and only if the other does; if one problem is feasible but unbounded, then the other is infeasible; if one is infeasible, then the other is either infeasible or feasible/unbounded. 3. Weak Duality Theorem: The objective function value of the primal (dual) to be minimized evaluated at any primal (dual) feasible solution cannot be less than the dual (primal) objective function value evaluated at a dual (primal) feasible solution. cTx >= bTy (in the standard equality form)
Relations between Primal and Dual (continued) 4. Strong Duality Theorem: When there is an optimal solution, the optimal objective value of the primal is the same as the optimal objective value of the dual. cTx* = bTy* 5. Complementary Slackness Theorem:Consider an inequality constraint (nonnegative variable) in any LP problem. If that constraint is inactive (nonnegative variable is positive) for an optimal solution to the problem, the corresponding dual variable (inequality constraint) will be zero (active) in any optimal solution for the dual of that problem. y*j (b-Ax*)j = 0, j=1,…,n; x*i (c-ATy*)i = 0, i=1,…,m;
World Cup Portfolio Management • Portfolio Management for World Cup Assets • Five securities in a market for open trading at fixed prices and pay-offs, and short is allowed • We’d like to decide how many shares to purchase or sell to maximize the pay-off when the game is realized.
Portfolio Optimization Model No share limit, and short is allowed:
Dual Portfolio Optimization Model P: the (normalized) state prices