Hard Optimization Problems: Practical Approach DORIT RON Tel Ziskind room #303

Course outline  1 st lecture: Introduction and motivation

INTRODUCTION  What is an optimization problem?

An optimization problem consist of:  Variables:  Energy functional to be minimized/maximized: min / max

Unconstrained minimization Find the global minimum

An optimization problem consist of:  Variables:  Energy functional to minimized/maximized: min / max Possibly subject to:  Equality constraints:  Inequality constraints:

Constrained minimization subject to

INTRODUCTION  What is an optimization problem?  Examples

Example 1: 2D Ising spins  Discrete (combinatorial) optimization min -   i,j> s i s j s i = { +1, -1}

3D Ising model  Each spin represents a tiny magnet  The spins tend to align below a certain T c  Ferromagnet – Iron at room temperature magnet  T c  non-magnet ----| | |--> T room temp ferromagnetism melting 770 o C 1538 o C  At T=0 the system settles at its ground states

Example 2: 1D graph ordering  Given a graph G=(V,  ), find a permutation  of the vertices that minimizes E(  )=  i j w i j |  (i) -  (j) | p where i, j are in V and w i j is the edge weight between i and j (w i j =0 if ij is not in  )  p=1 : Linear arrangement  p=2 : Quadratic energy  p= : The Bandwidth

i j Minimum Linear Arrangement Problem

i j

Minimum Linear Arrangement Problem i j

Minimum Linear Arrangement Problem i j

Minimum Linear Arrangement Problem i j

Minimum Linear Arrangement Problem i j

General Linear Arrangement Problem  Variable nodes sizes E(x)=  i j w i j | x i -x j | p x i = v i /2 +  k:  k)<  i) v k i j xixi xjxj

Other graph ordering problems  Minimize various functionals: envelope size, cutwidth, profile of graph, etc.  Traveling salesman problem – TSP

The Traveling Salesman Problem

Other graph ordering problems  Minimize various functionals: envelope size, cutwidth, profile of graph, etc.  Traveling salesman problem – TSP  Graph bisectioning  Graph partitioning  Graph coloring  Graph drawing

Drawing Graphs

Example 3: 2D circuit placement  Bottleneck in the microchip industry  Given a hypergraph, find the discrete placement of each module (gate) while minimizing the wirelength

The hypergraph for a microchip

Placement on a grid of pins

Routing over the placement

Example 3: 2D circuit placement  Bottleneck in the microchip industry  Given a hypergraph, find the discrete placement of each module (gate) while minimizing the wirelength  No overlap is allowed  No overflow is allowed  Critical paths must be shorter  Leave white space for routing  Typical IBM chip ~270 meters on 1cm 2

Place and route

Exponential growth of transistors for Intel processors

INTRODUCTION  What is an optimization problem?  Examples  Summary of difficulties

Difficulties:  Many variables: 10 6, 10 7 …  Many constraints: 10 6, 10 7 …  Multitude of local optima  Discrete nature  Conflicting objectives  Reasonable running time

INTRODUCTION  What is an optimization problem?  Examples  Summary of difficulties  Is the global optimum really needed / obtainable?

PEKO=PLACEMENT EXAMPLE WITH KNOWN OPTIMUM  Place the nodes – this is the solution  Create the net list locally and compactly  The optimum wire length – the sum of all the edges between the nodes, is known and can be proven to be minimal

SOLUTION QUALITY

INTRODUCTION  What is an optimization problem?  Examples  Summary of difficulties  Is the global optimum really needed / obtainable?  What is expected of a “good approximate” solution?

“Good approximate” solution  As optimal as possible: high quality solution  Achievable in linear time  Scalable in the problem size

RUNTIME

Reality Check Rigorous Optimization Theorems LIMITED Industrial Need for FAST & GOOD NP-Complete Intractable Problems HEURISTICS

INTRODUCTION  What is an optimization problem?  Examples  Summary of difficulties  Is the global optimum really needed / obtainable?  What is expected of a “good approximate” solution?  Multilevel philosophy

MULTILEVEL APPROACH  PARTIAL DIFFERENTIAL EQUATIONS (Achi Brandt since the early 70’s)  STATISTICAL PHYSICS  CHEMISTRY  IMAGE SEGMENTATION  TOMOGRAPHY  GRAPH OPTIMIZATION PROBLEMS

SOLUTION QUALITY

ORIGINAL PICTURE

ORIGINAL FENG SHUI (1) FENG SHUI (2) mPL KRAFTWERK CAPO DRAGON OURS

OUR PLACEMENT

Course outline  1 st lecture: Introduction and motivation  2 nd – 4 th : Local processing (relaxation) Quadratic minimization, Newton’s method, Steepest descent, Line search, Lagrange multipliers, Active set approach, Linear programming  4 th – 5 th : Global approaches Simulated annealing, Genetic algorithms, Spectral method  6 th : Classical geometric multigrid  7 th : Algebraic multilevel  8 th : Graph coarsening  9 th – 12 th : Advanced multilevel topics