Chapter 1. Introduction

1. Chapter 1. Introduction • Ex : Diet Problem Daily requirements : energy(2000kcal), protein(55g), calcium(800mg)

2. Formulation: Subject to

3. 선형계획법 문제 (LP problem)  목적함수(objective function) 우변상수(right hand side) Subject to • 제약식 (constraints) • 비음제약식(nonnegativity constraints). (없을 경우도 있음, unrestricted or free variable)

4. Unusual formulations • Cutting stock problem : Rolls of papers with width W to be cut into small pieces(finals). bi pieces of width wi, i =1, 2, …, m need to be produced. How to cut the rolls to meet demands while minimizing wastes? Subject to n denotes the total number of possible patterns which can be a very large number. aij = k if the number of i-th piece produced in the j-th pattern is k. (Chapter 13 참조)

5. ex) raws W=100 in., need 97 finals of width 45 in. 610 finals of width 36 in. 395 finals of width 31 in. 211 finals of width 14 in. Min x1 +x2 + x3 + + x37 • Note: • number of patterns grows fast as problem becomes large (generate columns when needed) • round down fractional optimal solution to LP to obtain integer solution • extension to 2-dimensional cutting stock (nesting problem), 3-D packing

6. Minimization of piecewise linear convex function  Subject to Subject to c3’x+d3 f c2’x+d2 c1’x+d1 x

7. ex)parallel processor scheduling problem There are m processors and n jobs to be processed on any one of the processors. aij: processing time of job j on processor i. Assign jobs to processors so that overall finish time (makespan) is minimized. Formulation as an integer programming problem minimize s. t.  minimize z s. t.

8. Linear programming relaxation of an integer programming problem is obtained by dropping the integrality requirement on the variables and considering only the linear constraints. • The optimal value of the linear programming relaxation provides a lower bound (for the minimization problem) on the optimal value of the integer programming problem. Hence it can be used importantly in the algorithm for integer programming problem. • Note that cannot be formulated as a linear programming problem. (maximizing a convex function)

9. Special case of piecewise linear objective function : separable piecewise linear objective function. function f: Rn R is separable if f(x) = f1(x1) + f2(x2) + … + fn(xn) If obj. fn. is nonlinear, but separable -> may approximate it by piecewise linear fn. (need some caution) c1 < c2 < c3 < c4 fi(xi) c4 c3 slope: ci c2 c1 xi a1 a2 a3 0 x1i x4i x3i x2i

10. Express xi in the constraints as xi x1i + x2i + x3i + x4i , where 0  x1i  a1, 0  x2i  a2 - a1 , 0  x3i  a3 - a2, 0  x4i In the objective function : min c1x1i + c2x2i + c3x3i + c4x4i Since we solve min problem, it is guaranteed that if we get xki > 0 in an optimal solution --> xji , j < k have values at their upper bounds.

11. 용어 • Feasible solution (가능해) : LP 의 제약식을 모두 만족하는 x Rn • Optimal solution (최적해) : LP의 가능해중에서 목적함수값을 최대화하는 해. • Optimal value (최적값) : 최적해의 목적함수 값 • Infeasible LP : 가능해를 갖지 않는 LP • Unbounded : 어떠한 유한한 값 M 에 대해서도 M보다 큰 목적함수 값을 주는 가능해가 있는 LP 의 경우 unbounded 라고 부른다. ex: 어떤 LP 이던지 다음의 3가지중 하나의 경우에 해당됨 • 유한한 최적값을 갖는 최적해가 존재 (여러 개의 최적해가 존재 가능) • Infeasible • Unbounded

12. 역사 • Solving systems of linear inequalities : Fourier, 1826, not efficient (Chapter 16) • Simplex method : G. B. Dantzig, 1947 • Ellipsoid method : L. G. Khachian, 1979 First polynomial time algorithm (theoretically efficient algorithm) for LP, practically not efficient • Interior point method : L. Karmarkar, 1984 polynomial time algorithm, many variations, practically good performance • Theory of LP provides important foundation for many other disciplines like integer programming, networks and graphs, nonlinear programming, etc .

13. Standard form Any LP problem can be expressed as the standard form • Minimization problem : solve max (-c)’x, then take negative of the optimal objective value. Optimal solution is the same • Equality constraint : ai’x=bi ai’x  bi,, (-ai)’x(-bi) • Unrestricted variable xj substitute xj byin the objective function and in the constraints where (Simplex method find solution with at most one of is positive) • Some people use min(max) c’x, Ax = b, x 0 as standard form. subject to

14. ex) • Formulation for absolute values (Assume cj  0 for variables involving absolute values) Subject to Subject to

15. express free xj = xj+ - xj-, xj+ 0, xj- 0 xj = xj+ + xj- , xj+ 0, xj- 0 want xj = xj+ , if xj 0 = xj- , if xj < 0 + xj- need xj+ xj- = 0 in an optimal solution, i.e. at most one is positive. If cj > 0, this is guaranteed to hold in an optimal solution. • Alternative formulation: subject to