A New Penalty-SQP Method

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1 Background and Motivation Illustration of Numerical Results Final Remarks Frank E. Curtis Informs Annual Meeting, October 2008

2 Background and Motivation Illustration of Numerical Results Final Remarks Outline Background and Motivation Illustration of Numerical Results Final Remarks

3 Background and Motivation Illustration of Numerical Results Final Remarks Sequential Quadratic Programming We solve the nonlinear optimization problem min f (x) s.t. b(x) 0 c(x) = 0, where f : R n R, b : R n R l and c : R n R m are smooth, via min f k + fk T d k + 1 d T 2 k (H k + H k )d k s.t. b k + b T k d k 0 c k + c T k d k = 0 and promote convergence with the l 1 penalty function φ(x; ρ) ρf (x) + l m max{0, b i (x)} + c j (x). i=1 j=1

4 Background and Motivation Illustration of Numerical Results Final Remarks Handling infeasible subproblems We have various options when a given subproblem is infeasible Feasibility restoration: replace the subproblem with min l m r i + (s j + t j ) + 1 d T H 2 k k d k i=1 j=1 s.t. b k + b T k d k r c k + c T k d k = s + t (... but what if we move away from optimality?) Constraint relaxation: replace the subproblem with ( ) l m min ρ f k + fk T d k + 1 d T 2 k H k d k + r i + (s j + t j ) + 1 d T H 2 k k d k s.t. b k + b T k d k r c k + c T k d k = s + t i=1 (... but what if we move away from feasibility?) j=1

5 Background and Motivation Illustration of Numerical Results Final Remarks Penalty-SQP Capabilities Sl 1 QP Global Fast Local Feasible Infeasible

6 Background and Motivation Illustration of Numerical Results Final Remarks Penalty-SQP Capabilities Sl 1 QP Global Fast Local Feasible Infeasible

7 Background and Motivation Illustration of Numerical Results Final Remarks Penalty-SQP Capabilities Sl 1 QP Global Fast Local Feasible Infeasible

8 Background and Motivation Illustration of Numerical Results Final Remarks Penalty-SQP Capabilities Sl 1 QP Feasible Infeasible Global Fast Local

9 Background and Motivation Illustration of Numerical Results Final Remarks Penalty-SQP Capabilities Sl 1 QP Feasible Infeasible Global Fast Local???

10 Background and Motivation Illustration of Numerical Results Final Remarks MINLP methods

11 Background and Motivation Illustration of Numerical Results Final Remarks MINLP methods

12 Background and Motivation Illustration of Numerical Results Final Remarks Summary There is a need for algorithms that converge quickly, regardless of whether the problem is feasible or infeasible Interior-point methods are known to perform poorly in infeasible cases, but active set methods seem promising Room for improvement in active set methods, too Feasibility restoration techniques are an option, but we prefer a single algorithm with no heuristics

13 Background and Motivation Illustration of Numerical Results Final Remarks A single algorithmic framework Our goal is to design a single optimization algorithm designed to solve min f (x) (NLP) s.t. b(x) 0 c(x) = 0 or, if (NLP) is infeasible, { (FEAS) min v(x) } l m max{0, b i (x)} + c j (x) i=1 j=1 We combine (NLP) and (FEAS) to define (P) {min ρf (x) + v(x)} where ρ 0 is a penalty parameter to be updated dynamically

14 Background and Motivation Illustration of Numerical Results Final Remarks Our method for step computation and acceptance We generate a step via min q k (d k ; ρ) ( ρ f k + f T ) k d k dt k H k d k + l i=1 { max 0, b i } k bi T k dk + which is equivalent to the quadratic subproblem m c j k + cj T dk k dk T H k d k j=1 (Q ρ) ( ) min ρ f k + fk T d k + 1 d T 2 k H k d k + s.t. b k + b T k d k r c k + c T k d k = s + t l m r i + (s j + t j ) + 1 d T H 2 k k d k i=1 j=1 and we measure progress with the exact penalty function φ(x; ρ) ρf (x) + v(x)

15 Background and Motivation Illustration of Numerical Results Final Remarks A Penalty-SQP framework Step 0. Initialize x 0 and set η (0, 1), τ (0, 1) and k 0 Step 1. If x k solves (NLP) or (FEAS), then stop Step 2. Compute a value for the penalty parameter, call it ρ k Step 3. Compute d k by solving (Q ρ) with ρ ρ k Step 4. Let α k be the first member of the sequence {1, τ, τ 2,...} s.t. φ(x k ; ρ k ) φ(x k + α k d k ; ρ k ) ηα k [q k (0; ρ k ) q k (d k ; ρ k )] Step 5. Update x k+1 x k + α k d k, go to Step 1

16 Background and Motivation Illustration of Numerical Results Final Remarks A Penalty-SQP framework Step 0. Initialize x 0 and set η (0, 1), τ (0, 1) and k 0 Step 1. If x k solves (NLP) or (FEAS), then stop Step 2. Compute a value for the penalty parameter, call it ρ k Step 3. Compute d k by solving (Q ρ) with ρ ρ k Step 4. Let α k be the first member of the sequence {1, τ, τ 2,...} s.t. φ(x k ; ρ k ) φ(x k + α k d k ; ρ k ) ηα k [q k (0; ρ k ) q k (d k ; ρ k )] Step 5. Update x k+1 x k + α k d k, go to Step 1

17 Background and Motivation Illustration of Numerical Results Final Remarks Steering Rules We extend the steering rules for setting ρ k 1 1. First, we guarantee first-order progress toward minimizing the feasibility violation measure v(x) during each iteration 1 Byrd, Nocedal, Waltz (2008) and Byrd, López-Calva, Nocedal (2008)

18 Background and Motivation Illustration of Numerical Results Final Remarks Steering Rules We extend the steering rules for setting ρ k 2. Second, we avoid settling on a penalty function with ρ too large by ensuring that progress in φ is proportional to the largest potential progress in feasibility

19 Background and Motivation Illustration of Numerical Results Final Remarks Penalty parameter update a. If the current iterate is feasible, then the algorithm should reduce ρ, if necessary, to ensure linearized feasibility; i.e., l k (d k ) l i=1 { } max 0, bk i bk i T dk + m c j k + cj T k d = 0 j=1 b. Otherwise, ρ should be set so that l k (0) l k (d k ) ɛ 1(l k (0) l k ( d k )) q k (0; ρ k ) q k (d k ; ρ k ) ɛ 2(q k (0; 0) q k ( d k ; 0)) where the reference direction is given by d k arg min (Q ρ) for ρ = 0

20 Background and Motivation Illustration of Numerical Results Final Remarks Penalty parameter update, extended a. If the current iterate is feasible, then the algorithm should reduce ρ, if necessary, to ensure linearized feasibility; i.e., l k (d k ) l i=1 { } max 0, bk i bk i T dk + m c j k + cj T k d = 0 j=1 b. Otherwise, ρ should be set so that l k (0) l k (d k ) ɛ 1(l k (0) l k ( d k )) q k (0; ρ k ) q k (d k ; ρ k ) ɛ 2(q k (0; 0) q k ( d k ; 0)) c. Finally, if ρ is decreased we ensure ρ k ɛ 3 KKT error for (FEAS) 2

21 Background and Motivation Illustration of Numerical Results Final Remarks Strategy for fast convergence Hitting a moving target: x k x ρ ˆx where x k kth iterate of the algorithm x ρ solution of penalty problem (P) ˆx infeasible stationary point of (NLP), solution of (FEAS) It has been shown 2 that, for some C, C > 0, x k+1 ˆx x k+1 x ρ + x ρ ˆx C x k x ρ 2 + O(ρ) C x k ˆx 2 + O(ρ), so convergence is quadratic if ρ x k ˆx 2 2 Byrd, Curtis, Nocedal (2008)

22 Background and Motivation Illustration of Numerical Results Final Remarks Strategy for fast convergence Hitting a moving target: x k x ρ ˆx where x k kth iterate of the algorithm x ρ solution of penalty problem (P) ˆx infeasible stationary point of (NLP), solution of (FEAS)

23 Background and Motivation Illustration of Numerical Results Final Remarks Strategy for fast convergence Hitting a moving target: x k x ρ ˆx where x k kth iterate of the algorithm x ρ solution of penalty problem (P) ˆx infeasible stationary point of (NLP), solution of (FEAS)

24 Background and Motivation Illustration of Numerical Results Final Remarks An infeasible problem with a strict minimizer of infeasibility

25 Background and Motivation Illustration of Numerical Results Final Remarks Problem batch as a MINLP Adding the constraint tl[1] 5 creates an infeasible problem:

26 Background and Motivation Illustration of Numerical Results Final Remarks Summary We have argued for the need of optimization algorithms that converge quickly, even on infeasible problems We have presented a penalty-sqp approach that transitions smoothly between solving an optimization problem and the corresponding feasibility problem The novel components of the algorithm are: 1. the quadratic model q k (it is quadratic even for ρ = 0) 2. an appropriate update for ρ to ensure superlinear convergence on infeasible problem instances We have illustrated that the algorithm performs well on both feasible and infeasible problem instances

27 Background and Motivation Illustration of Numerical Results Final Remarks Penalty-SQP Capabilities Now Thanks! Questions? Sl 1 QP Feasible Infeasible Global Fast Local

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