Algorithms for constrained local optimization

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Algorithms for constrained local optimization Fabio Schoen 2008 http://gol.

1 Algorithms for constrained local optimization Fabio Schoen Algorithms for constrained local optimization p.

2 Feasible direction methods Algorithms for constrained local optimization p.

3 Frank Wolfe method Let X: convex set. Consider the problem: min f(x) x X Let x k X choosing a feasible direction d k corresponds to choosing a point x X : d k = x x k. Steepest descent choice: min x X T f(x k )(x x k ) (a linear objective with convex constraints, usually easy to solve). Let ˆx k be an optimal solution of this problem. Algorithms for constrained local optimization p.

4 Frank Wolfe If T f(x k )(ˆx k x k ) = 0 then T f(x k )d 0 for every feasible direction d first order necessary conditions hold. Otherwise, letting d k = ˆx k x, this is a descent direction along which a step α k (0, 1] might be chosen according to Armijo s rule. Algorithms for constrained local optimization p.

5 Convergence of Frank-Wolfe method Under mild conditions the method converges to a point satisfying first order necessary conditions. However it is usually extremely slow (convergence may be sub linear) It might find applications in very large scale problems in which solving the sub-problem for direction determination is very easy (e.g. when X is a polytope). Algorithms for constrained local optimization p.

6 Gradient Projection methods Generic iteration: x k+1 = x k + α k ( x k x k ) where the direction d k = x k x k is obtained finding x k = [x k s k f(x k )] + where: s k R + and [ ] + represents projection over the feasible set. Algorithms for constrained local optimization p.

7 The method is slightly faster than Frank-Wolfe, with a linear convergence rate similar to that of (unconstrained) steepest descent. It might be applied if projection is relatively cheap, e.g. when the feasible set is a box. A point x k satisfies first order necessary conditions d T f(x k ) 0 iff x k = [x k s k f(x k )] + Algorithms for constrained local optimization p.

8 Lagrange Multiplier Algorithms Algorithms for constrained local optimization p.

9 Barrier Methods min f(x) g j (x) 0 j = 1,...,r A Barrier is a continuous function which tends to + whenever x approaches the boundary of the feasible region. Examples of barrier functions: B(x) = j log( g j (x)) logaritmic barrier B(x) = j 1 g j (x) invers barrier Algorithms for constrained local optimization p.

10 Barrier Method Let ε k 0 and x 0 strictly feasible, i.e. g j (x 0 ) < 0 j. Then let x k = arg min x R n(f(x) + ε kb(x)) Proposition: every limit point of {x k } is a global minimum of the constrained optimization problem Algorithms for constrained local optimization p. 1

11 Analysis of Barrier methods Special case: a single constraint (might be generalized) Let x be a limit point of {x k } (a global minimum). If KKT conditions hold, then there exists a unique λ 0: f( x) + λ g( x) = 0 (with λg( x) = 0. x k, solution of the barrier problem satisfies min f(x) + ε k B(x) g(x) < 0 f(x k ) + ε k B(x k ) = 0 Algorithms for constrained local optimization p. 1

12 ... If B(x) = φ(g(x)), In the limit, for k : f(x k ) + ε k φ (g(x k )) g(x k ) = 0 lim ε k φ (g(x k )) g(x k ) = λ g( x) if lim k g(x k ) < 0 φ (g(x k )) g(x k ) K (finite) and Kε k 0 if lim k g(x k ) = 0 (thanks to the unicity of Lagrange multipliers), λ = lim k ε k φ (g(x k )) Algorithms for constrained local optimization p. 1

13 Difficulties in Barrier Methods strong numeric instability: the condition number of the hessian matrix grows as ε k 0 need for an initial strictly feasible point x 0 (partial) remedy: ε k is very slowly decreased and the solution of the k + 1 th problem is obtained starting an unconstrained optimization from x k Algorithms for constrained local optimization p. 1

14 Example min(x 1) 2 + (y 1) 2 x + y 1 Logarithmic Barrier problem: min(x 1) 2 + (y 1) 2 ε k log(1 x y) x + y 1 < 0 Gradient: 2(x 1) + ε k 1 x y 2(y 1) + ε k 1 x y Stationary points x = y = 3 4 ± 1+ε k 4 (only the - solution is acceptable) Algorithms for constrained local optimization p. 1

15 Barrier methods and L.P. min c T x Ax = b x 0 Logarithmic Barrier on x 0: min c T x ε j log x j Ax = b x > 0 Algorithms for constrained local optimization p. 1

16 The central path The starting point is usually associated with ε = and is the unique solution of min j log x j Ax = b x > 0 The trajectory x(ε) of solutions to the barrier problem is called the central path and leads to an optimal solution of the LP. Algorithms for constrained local optimization p. 1

17 Penalty Methods Penalized problem: min f(x) + ρp(x) where ρ > 0 and P(x) 0 with P(x) = 0 if x is feasible. Example: min f(x) h i (x) = 0 i = 1,...,m A penalized problem might be: min f(x) + ρ i h i (x) 2 Algorithms for constrained local optimization p. 1

18 Convergence of the quadratic penalty me (for equality constrained problems): let P(x;ρ) = f(x) + ρ i h i (x) 2 Given ρ 0 > 0, x 0 R n, k = 0, let x k+1 = arg minp(x;ρ k ) (found with an iterative method initialized at x k ); let ρ k+1 > ρ k, k := k + 1. If x k+1 is a global minimizer of P and ρ k then every limit point of {x k } is a global optimum of the constrained problem. Algorithms for constrained local optimization p. 1

19 Exact penalties Exact penalties: there exists a penalty parameter value s.t. the optimal solution to the penalized problem is the optimal solution of the original one. l 1 penalty function: P 1 (x;ρ) = f(x) + ρ i h i (x) Algorithms for constrained local optimization p. 1

20 Exact penalties for inequality constrained problems: min f(x) h i (x) = 0 g j (x) 0 the penalized problem is P 1 (x;ρ) = f(x)ρ i h i (x) + ρ j max(0, g j (x)) Algorithms for constrained local optimization p. 2

21 Augmented Lagrangian method Given an equality constrained problem, reformulate it as: min f(x) ρ h(x) 2 h(x) = 0 The Lagrange function of this problem is called Augmented Lagrangian: L(x;λ) = f(x) ρ h(x) 2 + λ T h(x) Algorithms for constrained local optimization p. 2

22 Motivation min x f(x) ρ h(x) 2 + λ T h(x) x L ρ (x, λ) = f(x) + i λ i h(x) + ρh(x) h(x) = x L(x, λ) + ρh(x) h(x) 2 xxl ρ (x, λ) = 2 f(x) + i λ i 2 h(x) + ρh(x) 2 h(x) + ρ h(x) T h(x) = 2 xxl(x, λ) + ρh(x) 2 h(x) + ρ h(x) T h(x) Algorithms for constrained local optimization p. 2

23 motivation... Let (x,λ ) an optimal (primal and dual) solution. Necessarily: x L(x,λ ) = 0; moreover h(x ) = 0 thus x L ρ (x,λ ) = x L(x,λ ) + ρh(x ) h(x ) = 0 (x,λ ) is a stationary point for the augmented lagrangian. Algorithms for constrained local optimization p. 2

24 motivation... Observe that: 2 xxl ρ (x,λ) = 2 xxl(x,λ) + ρh(x) 2 h(x) + ρ h(x) T h(x) = 2 xxl(x,λ) + ρ h(x) T h(x) Assume that sufficient optimality conditions hold: v T 2 xxl(x,λ )v > 0 v : v T h(x ) = 0, Algorithms for constrained local optimization p. 2

25 ... Let v 0 : v T h(x )= 0. Then v T 2 xxl ρ (x,λ )v T = v T 2 xxl(x,λ )v T + ρv T h(x ) T h(x )v = v T 2 xxl(x,λ )v T > 0 Algorithms for constrained local optimization p. 2

26 ... Let v 0 : v T h(x ) 0. Then v T 2 xxl ρ (x,λ )v T = v T 2 xxl(x,λ )v T + ρv T h(x ) T h(x )v = v T 2 xxl(x,λ )v T + ρ(v T h(x )) 2 which might be negative. However ρ > 0: if ρ ρ v T 2 xxl ρ (x,λ )v T > 0. Thus, if ρ is large enough, the Hessian of the augmented lagrangian is positive definite and x is a (strict) local minimum of L ρ (,λ ) Algorithms for constrained local optimization p. 2

27 Inequality constraints min f(x) g(x) 0 Nonlinear transformation of inequalities into equalities: min x,s f(x) g j (x) + s 2 j = 0 j = 1,p Algorithms for constrained local optimization p. 2

28 Given the problem min f(x) h i (x) = 0 i = 1,m g j (x) 0 j = 1,p an Augmented Lagrangian problem might be defined as min L ρ (x,z;λ,µ) = min x,z f(x) + λt h(x) ρ h(x) 2 + j µ j (g j (x) + z 2 j) ρ j (g j (x) + z 2 j) 2 Algorithms for constrained local optimization p. 2

29 ... Consider minimization with respect to z variables: min z µ j (g j (x) + zj) ρ (g j (x) + zj) 2 2 j j µ j (g j (x) + u j ) ρ(g j(x) + u j ) 2 = min u 0 j (quadratic minimization over the nonnegative orthant). Solution: u j = max{0,ū j } where ū is the unconstrained optimum: ū : µ j + ρ(g j (x) + ū j ) = 0 Algorithms for constrained local optimization p. 2

30 ... Thus: u j = max{0, µ j ρ g j(x)}. Substituting: L ρ (x;λ,µ) = f(x) + λ T h(x) ρ h(x) ρ ( max{0,µj + ρg j (x)} µ j) 2 j This is an Augmented Lagragian for inequality constrained problems. Algorithms for constrained local optimization p. 3

31 Sequential Quadratic Programming minf(x) h i (x) = 0 Idea: apply Newton s method to solve the KKT equations: Lagrangian function: L(x;λ) = f(x) + i λ i h i (x) let H(x) = [h i (x)], H(x) = [ h i (x)]. KKT conditions: [ ] f(x) + H F[x;λ] = T (x)λ = 0 H(x) Algorithms for constrained local optimization p. 3

32 Newton step for SQP Jacobian of KKT system: F (x,λ) = [ 2 xx L(x;λ) T H(x) H(x) 0 ] Newton step: [ ] xk+1 λ k+1 = [ xk λ k ] + [ dk k ] where [ ][ ] 2 xx L(x k ;λ k ) T H(x k ) dk H(x k ) 0 k = [ f(xk ) H T (x k )λ k H(x k ) ] Algorithms for constrained local optimization p. 3

33 existence The Newton step exists if the Jacobian of the constraint set H(x k ) has full row rank the Hessian 2 xxl(x k ;λ k ) is positive definite In this case the Newton step is the unique solution of 2 xx L(x k;λ k )d k + T H(x k ) k + f(x k ) + H T (x k )λ k = 0 H(x k )d k + H(x k ) = 0 Algorithms for constrained local optimization p. 3

34 Alternative view: SQP KKT conditions: min d f(x k ) + f(x k ) T d dt 2 xxl(x k ;λ k )d H(x k )d + H(x k ) = 0 2 xxl(x k ;λ k )d + f(x k ) + H(x k )Λ k = 0 Under the same conditions as before this QP has a unique solution d k with Lagrange multipliers Λ k = λ k+1 Algorithms for constrained local optimization p. 3

35 Alternative view: SQP min d L(x k,λ k ) + T x L(x k,λ k )d dt 2 xxl(x k ;λ k )d KKT conditions: H(x k )d + H(x k ) = 0 2 xxl(x k ;λ k )d + f(x k ) + H(x k )λ k + H(x k )Λ k = 0 Under the same conditions as before this QP has a unique solution d k with Lagrange multipliers Λ k = k+1 Algorithms for constrained local optimization p. 3

36 Thus SQP can be seen as a method which minimizes a quadratic approximation to the Lagrangian subject to a first order approximation of the constraints. Algorithms for constrained local optimization p. 3

37 Inequalities If the original problem is then the SQP iteration solves min f(x) h i (x) = 0 g j (x) 0 min d f k + f(x k ) T d dt 2 xxl(x k,λ k )d T i h i (x k )p + h i (x k ) = 0 T j g j (x k )p + g j (x k ) 0 Algorithms for constrained local optimization p. 3

38 Filter Methods Basic idea: min f(x) g(x) 0 can be considered as a problem with two objectives: minimize f(x) minimize g(x) (the second objective has priority over the first) Algorithms for constrained local optimization p. 3

39 Filter Given the problem minf(x) g j (x) 0 j = 1,...,k let us consider the bi-criteria optimization problem min f(x) min h(x) where h(x) = j max{g j (x), 0} Algorithms for constrained local optimization p. 3

40 Let {f k,h k,k = 1, 2,...} the observed values of f and h at points x 1,x 2,... A pair (f k,h k ) dominates a pair (f l,h l ) iff f k f l and h k h l A filter is a list of pairs which are non-dominated by the others Algorithms for constrained local optimization p. 4

41 f(x) h(x) Algorithms for constrained local optimization p. 4

42 Trust region SQP Consider a Trust-region SQP method: min d f k + L(x k ;λ k ) T d dt 2 xxl(x k ;λ k )d T j g j (x k )p + g j (x k ) 0 d ρ (the norm is used here in order to keep the problem a QP) Traditional (unconstrained) trust region methods: if the current step is a failure reduce the trust region eventually the step will become a pure gradient step convergence! Algorithms for constrained local optimization p. 4

43 Trust region SQP Here diminishing the trust region radius might lead to infeasible QP s: T j g j(x k )p + g j (x k ) 0 g j (x) 0 x k Algorithms for constrained local optimization p. 4

44 Filter methods Data: x 0 : starting point, ρ, k = 0 while Convergence criterion not satisfied do if QP is infeasible then Find x k+1 minimizing constraint violation; else Solve QP and get a step d k ; try setting x k+1 = x k + d k ; end if (f k+1, h k+1 ) is acceptable to the filter then Accept x k+1 and add (f k+1, h k+1 ) to the filter; Remove dominated points from the filter; Possibly increase ρ; else Reject the step; Reduce ρ; end set k = k + 1; end Algorithms for constrained local optimization p. 4

45 Comparison with other methods f(x) Rejected filter steps acceptable steps "classical" method h(x) Algorithms for constrained local optimization p. 4

Lecture 13: Constrained optimization

Lecture 13: Constrained optimization 2010-12-03 Basic ideas A nonlinearly constrained problem must somehow be converted relaxed into a problem which we can solve (a linear/quadratic or unconstrained problem) We solve a sequence of such problems