A Trust-Funnel Algorithm for Nonlinear Programming

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1 for Nonlinear Programming Daniel P. Robinson Johns Hopins University Department of Applied Mathematics and Statistics Collaborators: Fran E. Curtis (Lehigh University) Nic I. M. Gould (Rutherford Appleton Laboratory) Philippe Toint (University of Namur) Southern California Optimization Day May 23, 2014 trust-funnel SoCalOptDay / 25

2 Outline 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

3 Outline 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

4 Outline 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

5 The aim of this tal Present an algorithm based on the trust-funnel concept for min x f(x) s.t. c(x) 0 By introducing slacs, we have the problem min x, s where c(x, s) := c(x) + s f(x) s.t. c(x, s) = 0, s 0 We consider solving a sequence of barrier subproblems min x, s f(x) µ ln([s] i ) =: f(x, s) s.t. c(x, s) = 0 The naive approach of applying the equality constrained algorithm will not wor because of the implicit constraint s > 0 In particular, we require fraction-to-the boundary constraints, a slac reset procedure, and variable scaling trust-funnel SoCalOptDay / 25

6 The barrier subproblem min x, s f(x) µ ln([s] i ) =: f(x, s) s.t. c(x, s) = 0 Basic subproblem: min d=(d x,d s ) mf (d) = f(x, s ) + f(x, s ) T d dt H d subject to c(x, s ) + J(x, s )d = 0 P 1 d 2 δ s + d s κ fb s where κ fb (0, 1), J(x, s) := c(x, s), ( ) ( ) I 0 2 xx L(x, y ) 0 P = and H 0 S := 0 Y S 1 trust-funnel SoCalOptDay / 25

7 Normal step computation Projected gradient step computation Tangential step computation y-iterations t C d = (x, s) + n + t t f-iterations r n (x, s) + n c + Jd = 0 v-iterations m f (n) (x, s) c + Jd 2 2 ν δ s 0 κfbs trust-funnel SoCalOptDay / 25

8 Outline The normal step 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

9 The normal step The normal step n The idea Aim to reduce v(x, s) = c(x, s) 2 How do we do this? By computing an approximate solution to min m v n (n) s.t. P 1 n δ v, s + n s κ fb s m v (n) = c(x, s ) + J(x, s )n 2 κ (0, 1) δ v > 0 is a trust-region radius Note: Assume that we now a value v max such that v(x, s ) v max trust-funnel SoCalOptDay / 25

10 The normal step The normal step n What do we mean by an approximate solution? We require that n satisfies P 1 n δ v s + n s κ fbs m v (n ) m v (nc ) P 1 n range ( P J(x, s ) T) (x, s) + n (x, s) n δ v c(x, s) + J(x, s)n = 0 How can we guarantee this? truncated preconditioned CG dogleg approach π v = P J(x, y ) T c(x, s ) Main diagram s 0 κfbs trust-funnel SoCalOptDay / 25

11 Outline The projected gradient step 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

12 The projected gradient step An approximate projected gradient r Define the approximate projected gradient as r = P 2 [ f m (n ) + J(x, s ) T ] y where y is an approximate solution to 1 min y R m 2 P [ f m (n ) + J(x, s ) T y ] 2 that satisfies at least one of 1 π f ɛ and v ɛ (approximate KKT) 2 π f 1 2 πv (do not compute a tangential step) 3 χ f 1 2 πf (descent direction) where π f = P [ f m (n ) + J(x, s ) T ] f y and χ = mf (n ) T r π f Main diagram trust-funnel SoCalOptDay / 25

13 Outline The tangential step 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

14 The tangential step Relaxed SQP tangential step t Define the Cauchy point ( t t C := tc (αc), where T tc (α) := C x ) (α) t Cs (α) and α C is the minimizer of T min m f ( n + t C(α)) α 0 s.t. P 1 ( n + t C(α)) min{δ v, δf } s + n s + tc s (α) κ fb (s + n s ) Then, t is a relaxed SQP tangential step if m f (n + t ) m f (n + t C ) s + n s + ts κ fb(s + n s ) ( ) r x := α r s = αr (1a) (1b) P 1 (n + t ) 2 min{δ v, δf } (1c) m v (n + t ) κ tg m v (0) + (1 κ tg)m v (n ) (1d) Main diagram trust-funnel SoCalOptDay / 25

15 The tangential step Very Relaxed SQP tangential step t Define the Cauchy point ( t t C = tc (αc), where T tc (α) := C x ) (α) t Cs (α) and α C is the minimizer of T min α 0 s.t. m f P 1 ( n + t C (α)) ( ) r x := α r s = αr ( n + t C (α)) min{δ v, δf, κ vv max s + n s + tc s (α) κ fb (s + n s ) Then, t is a very relaxed SQP tangential step if Main diagram } m f (n ) m f (n + t ) m f (n ) m f (n + t C ) s + n s + ts κ fb(s + n s ) (2a) (2b) P 1 (n + t ) min{δ v, δf, κ vv max } (2c) m v (n + t ) κ tt v max (2d) trust-funnel SoCalOptDay / 25

16 Outline Which steps to compute? 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

17 Which steps to compute? Normal step Have to compute it: π v > 1 2 πf 1 or v 0.9v max Option to compute: π v > 0 Do not compute: π v = 0 Projected gradient step Compute it: P 1 n 0.9 min{δ v, δf } Option to compute it otherwise. Tangential step Compute iff a projected gradient was computed and π f > 1 2 πv trust-funnel SoCalOptDay / 25

18 Outline 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

19 Three types of steps: y-iterations focus on better multiplier estimates f-iterations focus on reducing the barrier function f v-iterations focus on reducing infeasibility as measured by v trust-funnel SoCalOptDay / 25

20 Step computation diagram Definition of a y-iteration We classify the th iteration as a y-iteration if n = t = 0. Updates for a y-iteration w +1 w δ f +1 δf, v max +1 vmax δv +1 δv Notation: w = (x, s ) and w +1 = (x +1, s +1 ) trust-funnel SoCalOptDay / 25

21 Step computation diagram Definition of an f-iteration We classify the th iteration as an f-iteration if t 0 v(w + d ) v max (recall v(w ) v max ) m f (w ) m f (w + d ) 1 [ f 2 m (w + n ) m f (w + d ) ] Updates for an f-iteration δ v +1 δv, vmax +1 vmax If f(w ) f(w + d ) 1 2 else w +1 w + d perform a slac reset possibly increase δ f w +1 w decrease δ f [ m f (w ) m f (w + d ) ] then trust-funnel SoCalOptDay / 25

22 Step computation diagram Definition of a v-iteration The th iteration as a v-iteration if it is not a y- or an f-iteration. Updates for a v-iteration δ f +1 δf If n 0, m v (w ) m v (w + d ) 1 2[ m v (w ) m v (w + n ) ], and v(w ) v(w + d ) 1 2[ m v (w ) m v (w + d ) ] then w +1 w + d perform a slac reset possibly increase δ v decrease v max else w +1 w, v max +1 vmax, decrease δv trust-funnel SoCalOptDay / 25

23 Outline 1 The normal step The projected gradient step The tangential step Which steps to compute? trust-funnel SoCalOptDay / 25

24 Presented an inexact barrier-sqp algorithm for solving general nonlinear optimization problems based on a trust-funnel approach. Trial steps are composite steps formed from a normal step (designed to improve feasibility) and a tangential step (designed to decrease the barrier objective function). The method is matrix free, i.e., all conditions may be obtained via iterative methods. Subsets of core calculations are performed during each iteration based on appropriate criticality measures. Effective preconditioning is a challenge. Numerical results are in progress (part of GALAHAD) trust-funnel SoCalOptDay / 25

25 References Nicholas I. M. Gould and Ph. L. Toint, Nonlinear programming without a penalty function or a filter. Mathematical Programming, Nicholas I. M. Gould and Daniel P. Robinson and Ph. L. Toint, Corrigendum: Nonlinear programming without a penalty function or a filter. Mathematical Programming, trust-funnel SoCalOptDay / 25

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

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