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Cutting Plane and Subgradient Methods Cutting Plane and Subgradient Methods John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA October 12, 2009 Collaborators: Luc Basescu, Brian Borchers, Mohammad Oskoorouchi, Srini Ramaswamy, Kartik Sivaramakrishnan, Mike Todd Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 1 / 72 Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 2 / 72 Interior Point Cutting Plane and Column Generation Methods Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 3 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 4 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction The Traveling Salesman Problem TSP Webpage Robert Bosch Bill Cook, Georgia Tech Oberlin College Lower bounds determined using cutting planes and branch-and-cut. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 5 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction MIP Formulation Want to solve the integer programming problem T maxy2IRm b y subject to AT y ≤ c yi integer for i 2 I Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 6 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ≥ a0 y ≤ c0 x 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ≥ a0 y ≤ c0 x 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ; ≥ a0 y ≤ c0 x x0 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction LP Relaxation Relax integrality restriction: Dual problem: Primal problem: T T m n maxy2IR b y minx2IR ;x0 c x + c0x0 T s.t. A y ≤ c s.t. Ax + a0x0 = b T ; ≥ a0 y ≤ c0 x x0 0 Add a cutting plane to the dual problem Corresponds to column generation in the primal. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 7 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration fy 2 IRm : AT y ≤ cg PP PP PPP rrr P rrrrr H HH Soln to LP relaxation rrHrrrr H t HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration fy 2 IRm : AT y ≤ cg PP PP PPP rrr P Cutting plane T a1 y = c1 rrrrr H HH Soln to LP relaxation rrHrrrr H t HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T fy 2 IR : A y ≤ c; a1 y ≤ c1g PP PP PPP rrr P Soln to LP relaxation rrrrr t H HH rrHrrrr H HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T fy 2 IR : A y ≤ c; a1 y ≤ c1g Cutting plane PP PP aT y = c PPP2 2 rrr P Soln to LP relaxation rrrrr t H HH rrHrrrr H HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T T fy 2 IR : A y ≤ c; a1 y ≤ c1; a2 y ≤ b2g PP PP PPP rrr P rrrrr H Solution to LP relaxation HH rrHrrrr d H HH H rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods Introduction Cutting plane algorithm illustration m T T T fy 2 IR : A y ≤ c; a1 y ≤ c1; a2 y ≤ b2g PP PP PPP rrr P rrrrr H Solution to LP relaxation HH rrHrrrr d H Solution is integral, HH H so optimal to IP rr Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 8 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Outline 1 Interior Point Cutting Plane and Column Generation Methods Introduction MaxCut Interior point cutting plane methods Warm starting Theoretical results Stabilization 2 Cutting Surfaces for Semidefinite Programming Semidefinite Programming Relaxations of dual SDP Computational experience Theoretical results with conic cuts Condition number 3 Smoothing techniques and subgradient methods 4 Conclusions Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 9 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Valid constraints for MaxCut Find the maximum cut in a graph. Find ground state of Ising spin glass. Vertices $ spins: “Up” or “Down”. Edges $ interactions: ±1. t t t t t Interactions known. Deduce spins. t t t t t 1 u; v opposite spins y = t t t t t uv 0 u; v same spin t t t t t y $ incidence vector of cut. t t t t t Valid constraint: Any cut and any cycle intersect f e t tg in an even number of edges. h ye +yf +yg −yh ≤ 2 t t Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 10 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Valid constraints for MaxCut Find the maximum cut in a graph. Find ground state of Ising spin glass. Vertices $ spins: “Up” or “Down”. Edges $ interactions: ±1. t t t t t Interactions known. Deduce spins. t t t t t 1 u; v opposite spins y = t t t t t uv 0 u; v same spin t t t t t y $ incidence vector of cut. t t t t t Valid constraint: Any cut and any cycle intersect f e t tg in an even number of edges. h ye +yf +yg −yh ≤ 2 t t Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 10 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2. Can typically solve 100x100 grids in about 5 minutes with be = ±1. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 11 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2. Can typically solve 100x100 grids in about 5 minutes with be = ±1. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 11 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2. Can typically solve 100x100 grids in about 5 minutes with be = ±1. Mitchell (RPI) Cutting Planes, Subgradients INFORMS TutORial 11 / 72 Interior Point Cutting Plane and Column Generation Methods MaxCut Integer programming formulation P max e2E beye subject to y satisfies all cut-cycle inequalities (IP) y binary P max e2E beye subject to y satisfies all cut-cycle inequalities (LP) y satisfies additional linear constraints 0 ≤ y ≤ 1 Algorithm: 1 Initialize: just the box constraints 0 ≤ y ≤ 1. 2 Solve LP relaxation 3 Add violated constraints as required. 4 If not yet converged, return to Step 2.
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