Report

Introduction to Network Mathematics (1) - Optimization techniques Yuedong Xu 10/08/2012 Purpose • Many networking/system problems boil down to optimization problems – Bandwidth allocation – ISP traffic engineering – Route selection – Cache placement – Server sleep/wakeup – Wireless channel assignment – … and so on (numerous) Purpose • Optimization as a tool – shedding light on how good we can achieve – guiding design of distributed algorithms (other than pure heuristics) – providing a bottom-up approach to reverseengineer existing systems Outline • • • • • • Toy Examples Overview Convex Optimization Linear Programming Linear Integer Programming Summary Toys • Toy 1: Find x to minimize f(x) := x2 . x*=0 if no restriction on x . x*=2 if 2≤x≤4 Toys • Toy 2: Find the minimum minimize subject to x 0 , 10 i 1 log( x i a i ) Optimal solution ? 10 i 1 xi 1 Toys • Toy 3: Find global minimum local minimum global minimum Outline • • • • • Toy Examples Overview Convex Optimization Linear Programming Linear Integer Programming Overview Ingredients! • Objective Function – A function to be minimized or maximized • Unknowns or Variables – Affect value of objective function • Constraints – Restrict unknowns to take on certain values but exclude others Overview • Formulation: Minimize f0(x) subject to Objective function fi(x) ≤ 0; i=1,…,m Inequality constraint hi(x) = 0; i=1,…,p Equality constraint x: Decision variables Overview • Optimization tree Overview • Our coverage Nonlinear Programs Convex Programs Linear Programs (Polynomial) Flow and Matching Integer Programs (NP-Hard) Outline • • • • • Toy Examples Overview Convex Optimization Linear Programming Linear Integer Programming Convex Optimization • Concepts – Convex combination: x, y Rn 01 z = x +(1)y A convex combination of x, y. A strict convex combination of x, y if 0, 1. Convex Optimization • Concepts – Convex set: S n R convex is convex if it contains all convex combinations of pairs x, y S. nonconvex Convex Optimization • Concepts – Convex set: more complicated case The intersection of any number of convex sets is convex. Convex Optimization • Concepts – Convex function:(大陆教材定义与国外相反！) S Rn c:S R a convex set a convex function if c(x +(1)y) c(x) + (1)c(y), 0 1 c c(x) + (1)c(y) c(y) c(x) x c(x +(1)y) x +(1)y y Convex Optimization • Concepts – convex functions • Exponential: eax is convex on R • Powers: xa is convex on R+ when a≥1 or a≤0, and concave for 0≤a≤1 • Logarithm: log x is concave on R+ • Jensen’s inequality: – if f() is convex, then f(E[x]) <= E[f(x)] – You can also check it by taking 2nd order derivatives We are lucky to find that many networking problems have convex obj.s Convex Optimization • Concepts – Convex Optimization: Minimize f0(x) subject to fi(x) ≤ 0; i=1,…,m hi(x) = 0; i=1,…,p Convex Convex Linear/Affine – A fundamental property: local optimum = global optimum Convex Optimization • Method – You may have used • Gradient method • Newton method to find optima where constraints are real explicit ranges (e.g. 0≤x≤10, 0≤y≤20, -∞≤z≤∞ ……). – Today, we are talking about more generalized constrained optimization Convex Optimization How to solve a constrained optimization problem? • Enumeration? Maybe for small, finite feasible set • Use constraints to reduce number of variables? Works occasionally • Lagrange multiplier method – a general method (harder to understand) Convex Optimization Example: Minimize subject to x2 + y2 x+y=1 Lagrange multiplier method: • Change problem to an unconstrained problem: L(x, y, p) = x2 + y2 + p(1-x-y) • Think of p as “price”, (1-x-y) as amount of “violation” of the constraint • Minimize L(x,y,p) over all x and y, keeping p fixed • Obtain x*(p), y*(p) • Then choose p to make sure constraint met • Magically, x*(p*) and y*(p*) is the solution to the original problem! Convex Optimization Example: Minimize x2 + y2 subject to x + y = 1 Lagrangian: L(x, y, p) = x2 + y2 + p(1-x-y) • Setting dL/dx and dL/dy to 0, we get x = y = p/2 • Since x+y=1, we get p=1 • Get the same solution by substituting y=1-x Convex Optimization • General case: minimize f0(x) subject to fi(x) ≤ 0, i = 1, …, m (Ignore equality constraints for now) Optimal value denoted by f* Lagrangian: L(x, p) = f0(x) + p1f1(x) + … + pmfm(x) Define g(p) = infx (f0(x) + p1f1(x) + … + pmfm(x)) If pi≥0 for all i, and x feasible, then g(p) ≤ f* Convex Optimization • Revisit earlier example L(x,p) = x2 + y2 + p(1-x-y) x* = y* = p/2 g(p) = p(1-p/2) This is a concave function, with g(p*) = 1/2 • We know f* is 1/2, and g(p) is a lower bound for f* with different values of p – the tightest lower bound occurs at p=p*. Convex Optimization • Duality • For each p, g(p) gives a lower bound for f* • We want to find as tight a lower bound as possible: maximize g(p) subject to p≥0 a) This is called the (Lagrange) “dual” problem, original problem the “primal” problem b) Let the optimal value of dual problem be denoted d*. We always have d* ≤ f* (called “weak duality”) c) If d* = f*, then we have “strong duality”. The difference (f*-d*) is called the “duality gap” Convex Optimization • Price Interpretation • Solving the constrained problem is equivalent to obeying hard resource limits • Imaging the resource limits can be violated; you can pay a marginal price per unit amount of violation (or receive an amount if the limit not met) • When duality gap is nonzero, you can benefit from the 2nd scenario, even if the price are set in unfavorable terms Convex Optimization • Duality in algorithms An iterative algorithm produces at iteration j • A primal feasible x(j) • A dual feasible p(j) With f0(x(j))-g(p(j)) -> 0 as j -> infinity The optimal solution f* is in the interval [g(p(j)), f0(x(j))] Convex Optimization • Complementary Slackness To make duality gap zero, we need to have pi fi(x) = 0 for all i This means pi > 0 => fi(x*) = 0 fi(x*) < 0 => pi = 0 If “price” is positive, then the corresponding constraint is limiting If a constraint is not active, then the “price” must be zero Convex Optimization • KKT Optimality Conditions • Satisfy primal and dual constraints fi(x*) ≤0, pi* ≥0 • Satisfy complementary slackness pi* fi(x*) = 0 • Stationary at optimal f0’(x*) + Σi pi* fi’(x*) = 0 If primal problem is convex and KKT conditions met, then x* and p* are dual optimal with zero duality gap KKT = Karush-Kuhn-Tucker Convex Optimization • Method – Take home messages • Convexity is important (non-convex problems are very difficult) • Primal problem is not solved directly due to complicated constraints • Using Lagrangian dual approach to obtain dual function • KKT condition is used to guarantee the strong duality for convex problem Convex Optimization • Application: TCP Flow Control – What’s in your mind about TCP? • Socket programming? • Sliding window? • AIMD congestion control? • Retransmission? • Anything else? Convex Optimization • Application: TCP Flow Control – Our questions: • It seems that TCP works as Internet scales. Why? • How does TCP allocate bandwidth to flows traversing the same bottlenecks? • Is TCP optimal in terms of resource allocation? Convex Optimization • Application: TCP Flow Control Network Links l of capacities cl Sources i L(s) - links used by source i (routing) Ui(xi) - utility if source rate = xi i). The larger rate xi, the more happiness; ii). The increment of happiness is shrinking as xi increases. Convex Optimization • Application: TCP Flow Control – A simple network x1 x1 x 3 c 2 x1 x 2 c1 c1 c2 x2 x3 Convex Optimization • Application: TCP Flow Control – Primal problem max xi 0 subject to U (x ) i i i yl cl , l L Convex Optimization • Application: TCP Flow Control – Lagrangian dual problem Dual function : D ( x, p ) U s ( x s ) pl (cl x ) l s Dual problem : min p0 D ( p ) max xs 0 l l U ( x ) p ( c x s s l l ) s l You can solve it using KKT conditions! You need whole information! Convex Optimization • Application: TCP Flow Control – Question: Is distributed flow control possible without knowledge of the network? Convex Optimization • Application: TCP Flow Control – Primal-dual approach: • Source updates sending rate xi • Link generates congestion signal – Source’s action: • If congestion severe, then reduce xi • If no congestion, then increase xi – Congestion measures: • Packet loss (TCP Reno) • RTT (TCP Vegas) Convex Optimization • Application: TCP Flow Control – Gradient based Primal-dual algorithm: Source updates rate given congestion signal source : '1 x i ( t 1) U i ( q i ( t )) Total prices of a flow along its route Link updates congestion signal link : p l ( t 1) [ p l ( t ) l ( y l ( t ) c l )] Price of congestion at a link: packet drop prob., etc. Convex Optimization • Application: TCP Flow Control – Relation to real TCP Convex Optimization • Application: Fairness concerns – Many concepts regarding fairness • Max-min, proportional fairness, etc. • (add some examples to explain the concepts) Convex Optimization • Application: Fairness concerns – How does fairness relate to optimization? Reflected by utility function U s x ws x 1 1 U s x w s log x if 1, otherwise – Esp. max-min fairness as alpha infinity Convex Optimization • Application: TCP Flow Control Take home messages: – TCP is solving a distributed optimization problem! – Primal-dual algorithm can converge! – Packet drop, and RTT are carrying “price of congestion”! – Fairness reflected by utility function Outline • • • • • • Toy Examples Overview Convex Optimization Linear Programming Linear Integer Programming Summary Linear Programming • A subset of convex programming • Goal: maximize or minimize a linear function on a domain defined by linear inequalities and equations • Special properties Linear Programming • Formulation – General form Max or min c1 x1 Subject to a11 x1 a11 x1 c2 x2 a12 x 2 a12 x 2 cn xn a1 n x n a1 n x n b1 b1 amn xn amn xn b m b m …… a m 1 x1 a m 1 x1 am 2 x2 am 2 x2 Linear Programming • Formulation – General form Max or min cA1 xlinear c 2objective x2 function cn xn 1 Subject to a11 x1 a12 x 2 a1 n x n aA x1 of alinear x a1 n x n constraints 11 set 12 2 …… • linear inequalities or linear a• x aequations x m1 1 m2 2 a m 1 x1 a m 2 x 2 amn xn amn xn b1 b1 b m b m Linear Programming • Formulation – Canonical form Min subject to “greater than” only c2 x2 cn xn a11 x1 a12 x 2 a1 n x n b1 a 21 x1 a 22 x 2 a2 n xn b2 a m 1 x1 am 2 x2 amn xn bm c1 x1 Non-negative x1 , x 2 , , xn 0 Linear Programming • Example y Maximize 3x 2y Subject to x 3y 12 x y 8 2x y 10 x 0 y 0 8 7 6 x 12 y 3 x y 1 0 5 y 4 8 2x 3 2 1 x 1 2 3 4 5 6 7 8 Linear Programming • Example 3x 2y Subject to x 3y 12 x y 8 2x y 10 x 0 y 0 8 7 6 x 19 y y 2 8 y 2 15 y 2 3x 3x 1 2 2 3x 3 22 y 8 2 y x 3x 3x 4 y 1 0 3x 5 2 12 y 3 2x Maximize y 6 y 0 1 x 2 3 4 5 6 7 8 Linear Programming • Example – Observations: • Optimum is at the corner! – Question: • When can optimum be a point not at the corner? • How to find optimum in a more general LP problem? feasible region Linear Programming (Dantzig 1951) Simplex method •For USA Air Force •Very efficient in practice • Exponential time in worst case (Khachiyan 1979) Ellipsoid method • Not efficient in practice • Polynomial time in worst case Linear Programming • Simplex method – Iterative method to find optimum – Finding optimum by moving around the corner of the convex polytope in cost descent sense. basic feasible at a corner solution – Local optimum = global optimum (convexity) Linear Programming • Application – Max flow problem Outline • • • • • • Toy Examples Overview Convex Optimization Linear Programming Linear Integer Programming Summary Linear Integer Programming • Linear programming is powerful in many areas • But ……… Linear Integer Programming • Random variables are integers – Building a link between two nodes, 0 or 1? – Selecting a route, 0 or 1? – Assigning servers in a location – Caching a movie, 0 or 1? – Assigning a wireless channel, 0 or 1? – and so on…… No fractional solution ! Linear Integer Programming • Graphical interpretation Only discrete values allowed Linear Integer Programming • Method – Enumeration – Tree Search, Dynamic Programming etc. x1=0 X2=0 X2=1 X2=2 x1=2 x1=1 X2=0 X2=1 X2=2 X2=0 Guaranteed “feasible” solution Complexity grows exponentially X2=1 X2=2 Linear Integer Programming • Method – Linear relaxation plus rounding Integer Solution -cT x2 a). Variables being continuous b). Solving LP efficiently LP Solution c). Rounding the solution x1 No guarantee of gap to optimality! Linear Integer Programming • Combined Method – Branch and bound algorithm Step 1: Solve LP relaxation problem Step 2: Linear Integer Programming • Example – Knapsack problem Which boxes should be chosen to maximize the amount of money while still keeping the overall weight under 15 kg ? Linear Integer Programming • Example – Knapsack problem x1 {0,1} x 4 {0,1} Objective Function M axim ize 4 x1 2 x 2 10 x 3 2 x 4 1 x 5 x 2 {0,1} Unknowns or Variables x i 's, i 1, x 5 {0,1} x 3 {0,1} ,5 Constraints S ubject t o 12 x1 1 x 2 4 x 3 2 x 4 1 x 5 15 x i 0,1 , i 1, ,5 Linear Integer Programming • Example – Knapsack problem M axim ize 4 x1 2 x 2 10 x 3 2 x 4 1 x 5 S ubject to 12 x1 1 x 2 4 x 3 2 x 4 1 x 5 15, x i 0,1 , i 1, ,5 Outline • • • • • • Toy Examples Overview Convex Optimization Linear Programming Linear Integer Programming Summary Summary Our course covers • a macroscopic view of optimizations in networking • ways of solving optimizations • applications to networking research Summary Keywords • Convex optimization – Lagrangian dual problem, KKT • Linear programming – Simplex, Interior-point • Integer linear programming – Dynamic programming, Rounding, or Branch and bound Summary Complexity (“solve” means solving optimum) • Linear programming – [P], fast to solve • Nonlinear programming – Convex: [P], easy to solve – Non-convex: usually [NP], difficult to solve • (Mixed) Integer linear programming – Usually [NP], difficult to solve Summary Thanks! Backup slides Convex Optimization • Concepts – Convex function: more examples (see Byod’s book) We are lucky to find that many networking problems have convex obj.s Convex Optimization • Example – waterfilling problem To Tom: you can write The Lagrangian dual function On the white board. Convex Optimization • Method – Roadmap • Treat the original optimization problem as the Primal problem • Transform primal problem into dual problem • Solve dual problem and validate the optimality Convex Optimization • Method – Lagrangian dual function Convex Optimization • Method – For given ( , ) , solving dual problem Convex Optimization • Method – Comparison What shall we do next? Primal: Dual: Min f0(x) s.t. fi(x) ≤ 0; i=1,…,m hi(x) = 0; i=1,…,p Min L ( x , , ) If λ ≥ 0, dual minimum is lower bound of primal minimum Convex Optimization • Method – The Lagrangian dual problem Much easier after getting rid of ugly constraints! Convex Optimization • Method – Question: are these two problems the same? Convex Optimization • Method – Karush-Kuhn-Tucker (KKT) conditions For the convex problem, satisfying KKT means strong duality!