Optimization Problems

Optimization Problems Where we learn more fun things we can do with constraints. CSC5240 - Simplification, Optimization and Implication

Optimization - 1 • Often given some problem which is modelled by constraints we don’t want just any solution, but a “best” solution • This is an optimization problem • We need an objective function, a mapping from solutions to reals, so that we can rank solutions • By convention, we always try to minimize CSC5240 - Simplification, Optimization and Implication

Optimization - 2 • An optimization problem(C,f) consists of a constraint C and objective function f • A valuation v1 is preferred to valuation v2 if f(v1) < f(v2) • An optimal solution is a solution of C such that no other solution of C is preferred to it • Optimization problems may have more than one optimal solution CSC5240 - Simplification, Optimization and Implication

Optimization Example An optimization problem Find the closest point to the origin satisfying C. Some solutions and f values Optimal solution CSC5240 - Simplification, Optimization and Implication

Optimization Peculiarities • Some optimization problems have no solution. • Constraint has no solution • For any solution, there is a more preferable one. In other words, problem has no optimum. Can you give an example? CSC5240 - Simplification, Optimization and Implication

Optimization for FD CSPs • Domains are finite • We can use a solver to build a straightforward optimizer • retry_int_opt(C, D, f, best) • D2 := int_solv(C,D) • ifD2 is a false domain then returnbest • let sol be the solution corresponding to D2 • return retry_int_opt(C /\ f < sol(f), D, f, sol) CSC5240 - Finite Constraint Domains

Retry Optimization Example Smugglers knapsack problem (optimize profit) First solution found: Corresponding solution CSC5240 - Finite Constraint Domains

Retry Optimization Example Smugglers knapsack problem (optimize profit) Nextsolution found: Corresponding solution CSC5240 - Finite Constraint Domains

Retry Optimization Example Smugglers knapsack problem (optimize profit) No nextsolution! Corresponding solution CSC5240 - Finite Constraint Domains

Retry Optimization Example Smugglers knapsack problem (optimize profit) No nextsolution! Return best solution CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) • Since the solver may use backtrack search anyway combine it with the optimization • At each step in backtracking search, if best is the best solution so far add the constraint f < best(f) CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Current domain: CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Current domain: Initial bounds consistency CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Current domain: Initial bounds consistency W = 0 CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Current domain: Initial bounds consistency W = 0 P = 1 CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Current domain: Initial bounds consistency W = 0 Solution Found: add constraint P = 1 (0,1,3) CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Initial bounds consistency W = 0 P = 1 (0,1,3) CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Initial bounds consistency W = 0 P = 1 P = 2 (0,1,3) false CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Initial bounds consistency W = 0 P = 1 P = 2 P = 3 (0,1,3) false false CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Initial bounds consistency W = 0 W = 1 P = 1 P = 2 P = 3 (1,1,1) (0,1,3) false false CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Initial bounds consistency W = 0 W = 1 P = 1 P = 2 P = 3 (1,1,1) (0,1,3) false false Modify constraint CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Initial bounds consistency W = 0 W = 1 W = 2 P = 1 P = 2 P = 3 (1,1,1) false (0,1,3) false false Modify constraint CSC5240 - Finite Constraint Domains

Branch and Bound (BAB) Eg Smugglers knapsack problem (whiskey available) Initial bounds consistency W = 0 W = 1 W = 2 P = 1 P = 2 P = 3 (1,1,1) false (0,1,3) false false Return last sol (1,1,1) CSC5240 - Finite Constraint Domains

The Simplex Algorithm - 1 • The most widely used optimization algorithm • Optimizes a linear function wrt to linear constraints • Related to Gauss-Jordan elimination • Based on the following observations • Linear constraints form convex polygons • Optimal solution always occur at a vertex CSC5240 - Simplification, Optimization and Implication

The Simplex Form • An optimization problem (C, f) is in simplex form: • C is the conjunction of CE and CI • CE is a conjunction of linear equations • CI constrains all variables in C to be non-negative • f is a linear expression over variables in C CSC5240 - Simplification, Optimization and Implication

Simplex Form Example • An optimization problem in simplex form • An arbitrary problem can be put in simplex form by replacing • unconstrained var X by new vars and • inequality by new vars and CSC5240 - Simplification, Optimization and Implication

Simplex Solved Form - 1 • The simplex method works by repeatedly transforming an optimization problem in “basic feasible solved form” and to another • A simplex optimization problem is in basic feasible solved (bfs) form if: • The equations are in solved form • Each constant on the right hand side is non-negative • Only parameters occur in the objective CSC5240 - Simplification, Optimization and Implication

Simplex Solved Form - 2 • Every problem in bfs form has a corresponding “basic feasible solution” which corresponds to a vertex of the polygon defined by the constraint • A basic feasible solution is obtained by setting each parameter to 0 and each non-parameter to the constant in its equation CSC5240 - Simplification, Optimization and Implication

Simplex Solved Form Example • An equivalent problem to the previous example in the bfs form • We can read off a solution and its objective value CSC5240 - Simplification, Optimization and Implication

The Simplex Algorithm - 2 • The simplex algorithm finds the optimum by repeatedly looking for an “adjacent” basic feasible solved form whose basic feasible solution decreases the value of the objective function until no such neighbour can be found • By adjacent, we mean that the new bfs form can be reached by a single “pivoting” operation CSC5240 - Simplification, Optimization and Implication

The Simplex Algorithm - 3 starting from a problem in bfs form repeat Choose a variable y with negative coefficient in the obj. func. Find the equation x = b + cy + ... where c<0 and -b/c is minimal Rewrite this equation with y being the subject y = -b/c + 1/c x + ... Substitute -b/c + 1/c x + ... for y in all other eqns and obj. func. untilno such variable y exists or no such equation exists ifno such y exists then optimum is found elsethere is no optimum solution CSC5240 - Simplification, Optimization and Implication

Simplex Example Choose variable Z, the 2nd eqn is only one with neg. coeff Choose variable Y, the first eqn is only one with neg. coeff No variable can be chosen, optimal value 2 is found CSC5240 - Simplification, Optimization and Implication

Another Simplex Example - 1 An equivalent simplex form is: An optimization problem showing contours of the objective function CSC5240 - Simplification, Optimization and Implication

Another Simplex Example - 2 Basic feasible solved form: circle Choose S3, replace using 2nd eq Optimal solution: box CSC5240 - Simplification, Optimization and Implication

The Missing Part - 1 • How do we get the initial bfs form and its basic feasible solution? • Solve a different simplex problem • Add artificial variables to make equations in basic feasible solved form • Minimize the sum of the artificial variables • If the sum is zero, we can construct a bfs form for the original problem CSC5240 - Simplification, Optimization and Implication

The Missing Part Example - 1 Original simplex form equations With artificial vars in bfs form: Objective function rewritten: minimize CSC5240 - Simplification, Optimization and Implication

The Missing Part Example - 2 Problem after minimization of objective function Removing the artificial variables, the original problem is equivalent to CSC5240 - Simplification, Optimization and Implication

The Missing Part - 2 • Must remember to eliminate the basic variables of the resultant bfs form from the original objective function • Finding a bfs form of the original constraint is exactly a constraint satisfaction problem, and thus the first phase of the simplex method is an efficient constraint solver for linear inequalities CSC5240 - Simplification, Optimization and Implication

Integer Linear Programming • Unlike Simplex, the previous methods don’t use the objective function to direct search • Integer Linear Programming for (C,f) • use simplex to find a real optimal, • if solution is integer stop • otherwise choose a var x with non-integer opt value d and examine the problems • use the current best solution to constrain prob CSC5240 - Finite Constraint Domains

Integer Linear ProgrammingEg Smugglers knapsack problem CSC5240 - Finite Constraint Domains

Integer Linear Programming Eg Smugglers knapsack problem CSC5240 - Finite Constraint Domains

Optimization Problems