A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization with O(ɛ 3/2 ) Complexity

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Mohammadreza Samadi, Lehigh University joint work with

Curtis (stand-in presenter), Lehigh University OP17

1 A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization with O(ɛ 3/2 ) Complexity Mohammadreza Samadi, Lehigh University joint work with Frank E. Curtis (stand-in presenter), Lehigh University OP17 Vancouver, British Columbia, Canada 24 May 2017 A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 1 of 25

2 Outline Motivation Proposed Algorithm Theoretical Results Numerical Results Summary A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 2 of 25

3 Outline Motivation Proposed Algorithm Theoretical Results Numerical Results Summary A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 3 of 25

4 Introduction Consider nonconvex equality constrained optimization problems of the form min f(x) x R n s.t. c(x) = 0, where f : R n R and c : R n R m are twice continuously differentiable. We are interested in algorithm worst-case iteration / evaluation complexity. Constraints are not necessarily linear! (No projection-based algorithms.) A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 4 of 25

5 Algorithms Sequential Quadratic Optimization (SQP) / Newton s Method Trust Funnel; Gould & Toint (2010) Short-Step ARC; Cartis, Gould, & Toint (2013) A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 5 of 25

6 Algorithms Sequential Quadratic Optimization (SQP) / Newton s Method Global convergence: globally convergent (line search or trust region) Trust Funnel; Gould & Toint (2010) Global convergence: globally convergent Short-Step ARC; Cartis, Gould, & Toint (2013) Global convergence: globally convergent A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 5 of 25

7 Algorithms Sequential Quadratic Optimization (SQP) / Newton s Method Global convergence: globally convergent (line search or trust region) Worst-case complexity: No proved bound Trust Funnel; Gould & Toint (2010) Global convergence: globally convergent Worst-case complexity: No proved bound Short-Step ARC; Cartis, Gould, & Toint (2013) Global convergence: globally convergent Worst-case complexity: O(ɛ 3/2 ) (simplified; more later) A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 5 of 25

8 Trust region vs. ARC 1: Solve to compute s k : min s R n Trust Region q k(s) := f k + g T k s st H k s s.t. s 2 δ k (dual: λ k ) 2: Compute ratio: ρ q k f k f(x k +s k ) f k q k (s k ) 3: Update radius: ρ q k η: accept and δ k ρ q k < η: reject and δ k 1: Solve to compute s k : min s R n arc c k(s) := f k + g T k s st H k s σ k s 3 2 2: Compute ratio: ρ c k f k f(x k +s k ) f k c k (s k ) 3: Update regularization: ρ c k η: accept and σ k ρ c k < η: reject and σ k A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 6 of 25

9 Trust region vs. ARC: Subproblem solution correspondence 1: Solve to compute s k : min s R n Trust Region q k(s) := f k + g T k s st H k s s.t. s 2 δ k (dual: λ k ) 2: Compute ratio: ρ q k f k f(x k +s k ) f k q k (s k ) 3: Update radius: ρ q k η: accept and δ k ρ q k < η: reject and δ k σ k = λ k δ k δ k = s k 2 1: Solve to compute s k : min s R n arc c k(s) := f k + g T k s st H k s σ k s 3 2 2: Compute ratio: ρ c k f k f(x k +s k ) f k c k (s k ) 3: Update regularization: ρ c k η: accept and σ k ρ c k < η: reject and σ k A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 6 of 25

10 Short-Step ARC c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 7 of 25

11 Short-Step ARC c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 7 of 25

12 Short-Step ARC c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 7 of 25

13 Main concern Completely ignores the objective function during the first phase Question: Can we do better? A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 8 of 25

14 Contributions Completely ignores the objective function during the first phase Question: Can we do better? Yes!(?) First, consider TRACE method for unconstrained nonconvex optimization FEC, D. P. Robinson, M. Samadi, A trust region algorithm with a worst-case iteration complexity of O(ɛ 3/2 ) for nonconvex optimization, Mathematical Programming, 162(1 2), Second, rather than two-phase approach that ignores objective in phase 1, wrap in a trust funnel framework that observes objective in both phases. A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 8 of 25

15 Contributions Completely ignores the objective function during the first phase Question: Can we do better? Yes!(?) First, consider TRACE method for unconstrained nonconvex optimization FEC, D. P. Robinson, M. Samadi, A trust region algorithm with a worst-case iteration complexity of O(ɛ 3/2 ) for nonconvex optimization, Mathematical Programming, 162(1 2), Second, rather than two-phase approach that ignores objective in phase 1, wrap in a trust funnel framework that observes objective in both phases. Why trust funnel? Do not know, in general, how to bound number of updates to a penalty parameter and/or updates of filter entries! Trust funnel: driving factor is reducing constraint violation. A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 8 of 25

16 Outline Motivation Proposed Algorithm Theoretical Results Numerical Results Summary A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 9 of 25

17 SQP core Ideally, given x k, find s k as a solution of Issues: min f s R n k + gk T s st H k s s.t. c k + J k s = 0 H k (Hessian of Lagrangian) might not be positive definite over Null(J k ). Trust region!... but constraints might be incompatible. A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 10 of 25

18 Trust funnel basics Step decomposition approach: First, compute a normal step toward minimizing constraint violation: min v(x) = 1 2 c(x) 2 s 2 = norm R n mv k (snorm) s.t. s norm 2 δk v Second, compute multipliers λ k (or take from previous iteration). Third, compute a tangential step toward optimality: min s tang R n mf k (snorm + stang) s.t. J k s tang = 0, s norm + s tang 2 δ f k. A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 11 of 25

19 Tangential step s 2 c k + J k s = c k + J k s norm s 1 A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 12 of 25

20 Main idea Two-phase method combining trust funnel and trace. Trust funnel for globalization trace for good complexity bounds Phase 1 towards feasibility, two types of iterations: F-iterations improve objective and reduce constraint violation. V-iterations reduce constraint violation. Our algorithm vs. basic trust funnel modified F-iteration conditions and a different funnel updating procedure uses trace instead of tradition trust region ideas (for radius updates) after getting relatively feasible, switches to phase 2. A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 13 of 25

21 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

22 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

23 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

24 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

25 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

26 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

27 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

28 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

29 Illustration c(x) = 0 f = f A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 14 of 25

30 Outline Motivation Proposed Algorithm Theoretical Results Numerical Results Summary A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 15 of 25

31 Phase 1 Recall that v(x) = J(x) T c(x) and define the iteration index set Theorem 1 I := {k N : J T k c k 2 > ɛ v}. For any ɛ v (0, ), the cardinality of I is at most K(ɛ v), which accounts for O(ɛ 3/2 v ) successful steps and finite contraction and expansion steps between successful steps. Hence, the cardinality of I is K(ɛ v) = O(ɛ 3/2 v ). A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 16 of 25

32 Phase 1 Recall that v(x) = J(x) T c(x) and define the iteration index set Theorem 1 I := {k N : J T k c k 2 > ɛ v}. For any ɛ v (0, ), the cardinality of I is at most K(ɛ v), which accounts for O(ɛ 3/2 v ) successful steps and finite contraction and expansion steps between successful steps. Hence, the cardinality of I is K(ɛ v) = O(ɛ 3/2 v ). Corollary 2 If {J k } have full row rank with singular values bounded below by ζ (0, ), then has cardinality O(ɛ 3/2 v ). I c := {k N : c k 2 > ɛ v/ζ} A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 16 of 25

33 Phase 2 Options for phase 2: trust funnel method (no complexity guarantees) or target-following approach similar to Short-Step ARC to minimize Theorem 3 Φ(x, t) = c(x) f(x) t 2 For ɛ f (0, ɛ 1/3 v ], the number of iterations until is O(ɛ 3/2 f ɛ 1/2 v ). g k + J T k y k 2 ɛ f (y k, 1) 2 or J T k c k 2 ɛ f c k 2 Same complexity as Short-Step ARC: If ɛ f = ɛ 2/3 v, then overall O(ɛv 3/2 ) (though loose KKT tolerance). If ɛ f = ɛ v, then overall O(ɛ 2 v ). A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 17 of 25

34 Outline Motivation Proposed Algorithm Theoretical Results Numerical Results Summary A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 18 of 25

35 Implementation Matlab implementation: Phase 1: our algorithm vs. one doing V-iterations only Phase 2: trust funnel method [Curtis, Gould, Robinson, & Toint (2016)] Termination conditions: Phase 1: c k 10 6 max{ c 0, 1} or { J T k c k 10 6 max{ J T 0 c 0, 1} and c k > 10 3 max{ c 0, 1} Phase 2: g k + J T k y k 10 6 max{ g 0 + J T 0 y 0, 1}. A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 19 of 25

36 Test set Equality constrained problems (190) from the CUTEst test set: 78 constant (or null) objective 60 time limit (1 hour) 13 feasible initial point 3 infeasible phase 1 2 function evaluation error 1 small stepsizes (less than ) Remaining set consists of 33 problems. A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 20 of 25

37 TF TF-V-only Phase 1 Phase 2 Phase 1 Phase 2 Problem n m #V #F f g + J T y #V #F #V f g + J T y #V #F BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BT e e e e BYRDSPHR e e e e CHAIN e e e e FLT e e e e GENHS e e e e HS100LNP e e e e HS111LNP e e e e HS e e e e HS e e e e HS e e e e HS e e e e HS e e e e HS e e e e HS e e e e HS e e e e HS e e e e HS e e e e MARATOS e e e e MSS e e e e MWRIGHT e e e e ORTHREGB e e e e SPIN2OP e e time +1.67e e-01 time time A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 21 of 25

38 TF TF-V-only Phase 1 Phase 2 Phase 1 Phase 2 Problem n m #V #F f g + J T y #V #F #V f g + J T y #V #F BT e e e e BT e e e e A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 22 of 25

39 Summary of results Our algorithm, at the end of phase 1 for 26 problems, reaches a smaller function value for 6 problems, reaches the same function value Total number of iterations of our algorithm for 18 problems is smaller for 8 problems is equal A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 23 of 25

40 Outline Motivation Proposed Algorithm Theoretical Results Numerical Results Summary A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 24 of 25

41 Summary Proposed an algorithm for equality constrained optimization Trust funnel algorithm with improved complexity properties Promising performance in practice based on our preliminary experiment A step toward practical algorithms with good iteration complexity F. E. Curtis, D. P. Robinson, and M. Samadi. Complexity Analysis of a Trust Funnel Algorithm for Equality Constrained Optimization. Technical Report 16T-013, COR@L Laboratory, Department of ISE, Lehigh University, A Trust Funnel Algorithm for Nonconvex Equality Constrained Optimization 25 of 25

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