An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization

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Kristopher Poole
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Curtis, Lehigh University involving joint work with Travis Johnson,

Robinson, Johns Hopkins University Andreas Wächter, Northwestern

1 An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization Frank E. Curtis, Lehigh University involving joint work with Travis Johnson, Northwestern University Daniel P. Robinson, Johns Hopkins University Andreas Wächter, Northwestern University SIAM Conference on Optimization San Diego, CA 22 May 2014 An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 1 of 30

2 Outline Motivation Algorithm Description Numerical Experiments Summary An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 2 of 30

3 Problem formulation Our goal is to solve a large-scale constrained nonlinear optimization problem: min f(x) x R n s.t. c(x) = 0, c(x) 0. (NLP) If (NLP) is infeasible, then at least we want to minimize constraint violation: min v(x), where v(x) := c(x) 1 + [ c(x)] + 1. (FP) x R n Any feasible point for (NLP) is an optimal solution of (FP). An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 3 of 30

4 Algorithm design objectives 1 Demands on algorithms for large-scale constrained nonlinear optimization: Scalable step computation Effectively handles negative curvature Superlinear convergence in primal-dual space Asymptotic monotonicity (consistent progress toward solution) Active-set detection and warm-starting Traditional methods (SQO, IP, AL) only satisfy subsets of these demands. I can solve my problem quickly enough with an IP method using matrix factorizations... then you re not interested in what I have to say. If you can only afford a few NLP iterations, are solving sequences of similar problems, and/or need a method that can converge fast locally,... 1 Paraphrased from Zavala and Anitescu (2014) An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 4 of 30

5 Sequential quadratic optimization (SQO) SQO advantages: Parameter free search direction computation (ideally) Strong global convergence properties and behavior Active-set identification = Newton-like local convergence SQO disadvantages: No best way to handle negative curvature/inconsistent subproblems Quadratic subproblems (QPs) are expensive to solve exactly not scalable! Questions Employing scalable QP solvers, can we exploit inexact solves? From inexactness, do we lose global/local convergence? active set detection? Is there any benefit in applying a scalable method for the QP subproblems, as opposed to applying a similar scalable method to (NLP) directly? Sit and solve QP? Or move on nonlinear problem as in filtersd; Fletcher (2012) An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 5 of 30

6 Inexact SQO vs. inexact SQO Inexact SQO may mean inexact gradients with (in)exact QP solves: Dennis, El-Alem, Maciel (1997) Diehl, Walther, Bock, Kostina (2010) Jäger, Sachs (1997) Heinkenschloss, Vicente (2000) Walther, Vetukuri, Biegler (2012) Inexact SQO may also mean exact gradients with inexact QP solves: Byrd, Curtis, Nocedal (2008) Izmailov, Solodov (2010) Leibfritz, Sachs (1999) Morales, Nocedal, Wu (2010) In this talk, we are interested in the latter type of inexact SQO. An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 6 of 30

7 Algorithmic framework: Classic NLP solver approximation model solution QP solver approximation model solution linear solver An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 7 of 30

8 Algorithmic framework: Inexact NLP solver approximation model termination conditions approximate solution step type QP solver approximation model termination conditions approximate solution step type linear solver An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 8 of 30

9 SQO w/ inexact subproblem solves Contributions: Broad, but consistent, framework; Gill, Robinson (2013), pdal/ssqo Implementable termination conditions for inexact QP solves No specific QP solver required Global convergence guarantees (feasible and infeasible problems) Future work: Fast local convergence (feasible and infeasible problems) 2 Algorithmic features: Allows generic inexactness in QP solutions Convex combination of optimality and feasibility steps Negative curvature handled with dynamic Hessian modifications Separate multipliers for (NLP) and (FP) Dynamic updates for penalty parameter and Lagrange multipliers 2 Avoid using Cauchy points that only yield minimal progress for global convergence. An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 9 of 30

10 Fritz John and penalty functions (NLP): min f(x) x s.t. c(x) = 0, c(x) 0 (FP):» + min v(x) := c(x) x [ c(x)] 1 Define the Fritz John (FJ) function F(x, y, ȳ, µ) := µf(x) + c(x) T y + c(x) T ȳ and the l 1 -norm exact penalty function φ(x, µ) := µf(x) + v(x). µ 0 acts as objective multiplier/penalty parameter. An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 10 of 30

11 Optimality conditions (NLP): min f(x) x s.t. c(x) = 0, c(x) 0 (FP):» + min v(x) := c(x) x [ c(x)] 1 (PP): min x (FJ): φ(x, µ) := µf(x)+v(x) F(x, y, ȳ, µ) := µf(x) + c(x) T y + c(x) T ȳ KKT conditions for (FP) and (PP) expressed with residual 2 µg(x) + J(x)y + J(x)ȳ 3 min{[c(x)] +, e y} ρ(x, y, ȳ, µ) := 6 min{[c(x)], e + y} 7 4 min{[ c(x)] +, e ȳ} 5 min{[ c(x)], ȳ} FJ point: KKT point: ρ(x, y, ȳ, µ) = 0, v(x) = 0, (y, ȳ, µ) 0 ρ(x, y, ȳ, µ) = 0, v(x) = 0, µ > 0 Infeasible stationary point: ρ(x, y, ȳ, 0) = 0, v(x) > 0 An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 11 of 30

12 Penalty function model and QP subproblem (NLP): min f(x) x s.t. c(x) = 0, c(x) 0 (FP):» + min v(x) := c(x) x [ c(x)] 1 (PP): min x (FJ): φ(x, µ) := µf(x)+v(x) F(x, y, ȳ, µ) := Define a local model of φ(, µ) at x k : l k (d, µ) := µ(f k + g T k d) + c k + J T k d 1 + [ c k + J T k d]+ 1 Reduction in this model yielded by a given d: Subproblem of interest: l k (d, µ) := l(0, µ) l(d, µ) min d l k (d, µ) dt Hd (QP) µf(x) + c(x) T y + c(x) T ȳ KKT residual: ρ(x, y, ȳ, µ) l k (d, µ) > 0 implies d is a direction of strict descent for φ(, µ) from x k An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 12 of 30

13 Optimality conditions (for QP) (NLP): min f(x) x s.t. c(x) = 0, c(x) 0 (FP):» + min v(x) := c(x) x [ c(x)] 1 (PP): min x (FJ): φ(x, µ) := µf(x)+v(x) F(x, y, ȳ, µ) := µf(x) + c(x) T y + c(x) T ȳ KKT residual: ρ(x, y, ȳ, µ) Local model of φ at x k : l k (d, µ) KKT conditions for (FP) and (PP) expressed with residual 2 µg(x) + J(x)y + J(x)ȳ 3 min{[c(x)] +, e y} ρ(x, y, ȳ, µ) := 6 min{[c(x)], e + y} 7 4 min{[ c(x)] +, e ȳ} 5 min{[ c(x)], ȳ} KKT conditions for (QP) expressed with 2 µg k + Hd + J k y + J 3 k ȳ min{[c k + Jk T d]+, e y} ρ k (d, y, ȳ, µ, H) := min{[c 6 k + Jk T d], e + y} 4min{[ c k + J T 7 k d]+, e ȳ} 5 min{[ c k + J T k d], ȳ} Exact solution of (QP): ρ k (d, y, ȳ, µ, H) = 0 An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 13 of 30

14 Assumptions and well-posedness Assumption (1) The functions f, c, and c and their first derivatives are bounded and Lipschitz continuous in an open convex set containing {x k } and {x k + d k }. (2) The QP solver can solve (QP) arbitrarily accurately for any µ 0. Theorem (Well-posedness) One of the following holds for our method, isqo: 1. isqo terminates finitely with a KKT point or infeasible stationary point. 2. isqo generates an infinite sequence of iterates»» y x k, k y, k y ȳ k ȳ k, µ k where k y k [ e, e],»ȳ k [0, e], and µ k > 0. ȳ k An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 14 of 30

15 Global convergence Theorem (Global convergence) One of the following holds: (a) µ k = µ for some µ > 0 for all large k and either every limit point of {x k } corresponds to a KKT point or is an infeasible stationary point; (b) µ k 0 and every limit point of {x k } is an infeasible stationary point; (c) µ k 0, all limit points of {x k } are feasible, and, with K µ := {k : µ k+1 < µ k }, every limit point of {x k } k Kµ corresponds to an FJ point where the MFCQ fails. Corollary If {x k } is bounded and every limit point of this sequence is a feasible point at which the MFCQ holds, then µ k = µ for some µ > 0 for all large k and every limit point of {x k } corresponds to a KKT point. An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 15 of 30

16 Direct scenario (NLP): min f(x) x s.t. c(x) = 0, c(x) 0 (FP):» + min v(x) := c(x) x [ c(x)] 1 (PP): min x (FJ): φ(x, µ) := µf(x)+v(x) F(x, y, ȳ, µ) := µf(x) + c(x) T y + c(x) T ȳ KKT residuals: ρ(x, y, ȳ, µ) ρ k (d, y, ȳ, µ, H) Local model of φ at x k : l k (d, µ) Terminate the QP solver when the solution (d k, y k+1, ȳ k+1 ) of (QP) with µ = µ k satisfies If y k+1 [ e, e], ȳ k+1 [0, e] l k (d k, µ k ) θ d k 2 > 0 for θ (0, 1) ρ k (d k, y k+1, ȳ k+1, µ k, H k ) κ ρ(x k, y k, ȳ k, µ k ) l k (d k, µ k ) ɛv k for ɛ (0, 1) then d k d k is the search direction µ k+1 µ k An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 16 of 30

17 Reference scenario (NLP): min f(x) x s.t. c(x) = 0, c(x) 0 (FP):» + min v(x) := c(x) x [ c(x)] 1 (PP): min x (FJ): φ(x, µ) := µf(x)+v(x) F(x, y, ȳ, µ) := µf(x) + c(x) T y + c(x) T ȳ KKT residuals: ρ(x, y, ȳ, µ) ρ k (d, y, ȳ, µ, H) Local model of φ at x k : l k (d, µ) Terminate the QP solver when the solution (d k, y k+1, ȳ k+1 ) of (QP) with µ = 0 satisfies If y k+1 [ e, e], ȳ k+1 [0, e] l k (d k, 0) θ d k 2 for θ (0, 1) ρ k (d k, y k+1, ȳ k+1, 0, H k ) κ ρ(x k, y k, ȳ k, 0) l k (d k, µ k ) ɛ l k (d k, 0) for ɛ (0, 1) then d k d k is the search direction µ k+1 µ k An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 17 of 30

18 Combination scenario (NLP): min f(x) x s.t. c(x) = 0, c(x) 0 (FP):» + min v(x) := c(x) x [ c(x)] 1 (PP): min x (FJ): φ(x, µ) := µf(x)+v(x) F(x, y, ȳ, µ) := Choose the largest τ [0, 1] such that d k τd k + (1 τ)d k yields l k (d k, 0) ɛ l k (d k, 0) then choose µ k+1 < µ k such that l k (d k, µ k+1 ) β l k (d k, 0) for β (0, 1) µf(x) + c(x) T y + c(x) T ȳ KKT residuals: ρ(x, y, ȳ, µ) ρ k (d, y, ȳ, µ, H) Local model of φ at x k : l k (d, µ) An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 18 of 30

19 isqo framework repeat (1) Check whether KKT point or infeasible stationary point has been obtained. (2) Compute an inexact solution of (QP) with µ = µ k. (a) If Direct scenario occurs, then go to step 4. (3) Compute an inexact solution of (QP) with µ = 0. (a) If Reference scenario occurs, then go to step 4. (b) If Combination scenario occurs, then go to step 4. (4) Perform a backtracking line search to reduce φ(, µ k+1 ). endrepeat An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 19 of 30

20 Complicating factors A few special cases make our actual algorithm slightly ;-) more complicated Landing on stationary points for φ(, µ k ) We allow only a multiplier and/or penalty parameter update A tightened accuracy tolerance is needed in combination scenarios We may require certain multipliers to be close to their bounds (Think of identifying violated constraints) H k and/or H k may not be positive definite We ask the QP solver to check the curvature along trial directions (Dynamic inertia correction if trial curvature is too small/negative) Actual algorithm involves six scenarios, but we have presented the core ideas An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 20 of 30

21 Implementation details: Experiments in paper Matlab implementation Test set involves 309 CUTEr problems with at least one free variable at least one general (non-bound) constraint #variables + #constraints 20,000 (because it s Matlab!) no failures due to BQPD BQPD for QP solves with indefinite Hessians; see (Fletcher, 2000) Simulated inexactness by perturbing QP solutions Termination conditions (ɛ tol = 10 6 and ɛ µ = 10 8 ): ρ(x k, y k, ȳ k, µ k ) ɛ tol and v k ɛ tol ; (Optimal) ρ(x k, y k, ȳ k, 0) = 0 and v k > 0; (Infeasible) ρ(x k, y k, ȳ k, 0) ɛ tol and v k > ɛ tol and µ k ɛ µ (Infeasible) Investigate performance of inexact algorithm with κ = 0.01, 0.1, and 0.5 An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 21 of 30

22 Success statistics Counts of termination messages for exact and three variants of inexact algorithm: Termination message Exact Inexact κ = 0.01 κ = 0.1 κ = 0.5 Optimal solution found Infeasible stationary point found Iteration limit reached Termination statistics and reliability do not degrade with inexactness! An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 22 of 30

23 Inexactness levels Observe induced relative residuals for QP solves: κ I := ρ k ρ For problem j, we compute minimum (κ I (j)) and mean ( κ I (j)) values over run: min κ κ I,min κi(j) [ ) [ ) [ ) [ ) [ ) [ ) [ ) e e e mean κ κ I,mean e e e κi(j) [0.5 1) [1 ) Relative residuals generally need only be moderately smaller than parameter κ! An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 23 of 30

24 Iteration comparison Considering the logarithmic outperforming factor r j := log 2 (iter j inexact /iterj exact ), we compare iteration counts of our inexact (κ = 0.01) and exact algorithms: Iteration counts do not degrade significantly with inexactness! An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 24 of 30

25 Implementation details: Experiments from last night Matlab implementation Test set involves 60 CUTEr problems that were all successfully solved with augmented Lagrangian (AL) method sequential quadratic optimization (SQO) method isqo method with AL method for QP subproblems AL method or CPLEX for QP solves: switch when ρ(x k, y k, ȳ k, µ k ) 10 2 and v k 10 2 Terminate solve for (NLP) when ρ(x k, y k, ȳ k, µ k ) 10 6 and v k 10 6 An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 25 of 30

26 Iteration comparison: AL vs. SQO AL (left) with cheap iterations vs. SQO (right) with expensive iterations An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 26 of 30

27 Iteration comparison: SQO vs. isqo SQO (left) with expensive iterations vs. isqo (right) with cheaper iterations An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 27 of 30

28 Iteration comparison: AL vs. isqo(subproblems) AL (left) with cheap iterations vs. isqo (right) with few expensive iterations An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 28 of 30

29 Summary Contributions: Developed, analyzed, and experimented with an inexact SQO method Allows generic inexactness in QP subproblem solves No specific QP solver required Global convergence guarantees established Numerical experiments suggest inexact algorithm is reliable Inexact solutions allowed without degradation of performance An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 29 of 30

30 Thanks! Exact Algorithms: J. V. Burke, F. E. Curtis, and H. Wang, A Sequential Quadratic Optimization Algorithm with Rapid Infeasibility Detection, to appear in SIAM Journal on Optimization, R. H. Byrd, F. E. Curtis, and J. Nocedal, Infeasibility Detection and SQP Methods for Nonlinear Optimization, SIAM Journal on Optimization, Volume 20, Issue 5, pg , Inexact Algorithms: F. E. Curtis, T. C. Johnson, D. P. Robinson, and A. Wächter, An Inexact Sequential Quadratic Optimization Algorithm for Large-Scale Nonlinear Optimization, to appear in SIAM Journal on Optimization, F. E. Curtis, J. Huber, O. Schenk, and A. Wächter, A Note on the Implementation of an Interior-Point Algorithm for Nonlinear Optimization with Inexact Step Computations, Mathematical Programming, Series B, Volume 136, Issue 1, pg , F. E. Curtis, O. Schenk, and A. Wächter, An Interior-Point Algorithm for Large-Scale Nonlinear Optimization with Inexact Step Computations, SIAM Journal on Scientific Computing, Volume 32, Issue 6, pg , R. H. Byrd, F. E. Curtis, and J. Nocedal, An Inexact Newton Method for Nonconvex Equality Constrained Optimization, Mathematical Programming, Volume 122, Issue 2, pg , F. E. Curtis, J. Nocedal, and A. Wächter, A Matrix-free Algorithm for Equality Constrained Optimization Problems with Rank-Deficient Jacobians, SIAM Journal on Optimization, Volume 20, Issue 3, pg , R. H. Byrd, F. E. Curtis, and J. Nocedal, An Inexact SQP Method for Equality Constrained Optimization, SIAM Journal on Optimization, Volume 19, Issue 1, pg , An Inexact Sequential Quadratic Optimization Method for Nonlinear Optimization 30 of 30

Infeasibility Detection and an Inexact Active-Set Method for Large-Scale Nonlinear Optimization

Infeasibility Detection and an Inexact Active-Set Method for Large-Scale Nonlinear Optimization Frank E. Curtis, Lehigh University involving joint work with James V. Burke, University of Washington Daniel