Survey of NLP Algorithms. L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA

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1 Survey of NLP Algorithms L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA

2 NLP Algorithms - Outline Problem and Goals KKT Conditions and Variable Classification Handling Basic Variables Handling Nonbasic Variables Handling Superbasic Variables Ensuring Global Convergence Algorithms and Software Algorithmic Comparison Summary and Future Wor 2

3 Constrained Optimization (Nonlinear Programming) Problem: Min x f(x) s.t. g(x) h(x) = where: f(x) - scalar objective function x - n vector of variables g(x) - inequality constraints, m vector h(x) - meq equality constraints. Sufficient Condition for Unique Optimum - f(x) must be convex, and - feasible region must be convex, i.e. g(x) are all convex h(x) are all linear Except in special cases, there is no guarantee that a local optimum is global if sufficient conditions are violated. 3

4 Variable Classification for NLPs Min f(x) Min f(z) s.t. g(x) + s = (add slac variable) s.t. c(z) = h(x) =, s a z b Partition variables into: z B - dependent or basic variables z N - nonbasic variables, fixed at a bound z S - independent or superbasic variables Analogy to linear programming. Superbasics required only if nonlinear problem 4

5 Characterization of Constrained Optima (two primal, two slacs) Min Min Linear Program Two nonbasic Linear Program (Alternate Optima) nonbasic variables, two basic, no superbasic Min One nonbasic variable Two basic, one superbasic Min Min No nonbasic variables Two basic, two superbasic Convex Objective Functions Linear Constraints 5

6 What conditions characterize an optimal solution? Unconstrained Local Minimum Necessary Conditions f (x*) = p T 2 f (x*) p for p R n (positive semi-definite) Unconstrained Local Minimum Sufficient Conditions f (x*) = p T 2 f (x*) p > for p R n (positive definite) 6

7 Optimal solution for inequality constrained problem Min f(x) s.t. g(x) Analogy: Ball rolling down valley pinned by fence Note: Balance of forces ( f, g1) 7

8 Optimal solution for general constrained problem Problem: Min f(x) s.t. g(x) h(x) = Analogy: Ball rolling on rail pinned by fences Balance of forces: f, g1, h 8

9 Optimality conditions for local optimum Necessary First Order Karush Kuhn - Tucer Conditions L (z*, u, v) = f(x*) + g(x*) u + h(x*) v = (Balance Forces) u (Inequalities act in only one direction) g (x*), h (x*) = (Feasibility) u j g j (x*) = (Complementarity: either g j (x*) = or u j = ) u, v are "weights" for "forces," nown as KKT multipliers, shadow prices, dual variables To guarantee that a local NLP solution satisfies KKT conditions, a constraint qualification is required. E.g., the Linear Independence Constraint Qualification (LICQ) requires active constraint gradients, [ g A (x*) h(x*)], to be linearly independent. Also, under LICQ, KKT multipliers are uniquely determined. Necessary (Sufficient) Second Order Conditions - Positive curvature in "constraint" directions. - p T 2 L (x*) p (p T 2 L (x*) p > ) where p are the constrained directions: g A (x*) T p=, h(x*) T p= This is the space of the superbasic variables! 9

10 Optimality conditions for local optimum Min f(z), s.t. c(z) =, a z b L (z*, λ, ν) = f(z*) + c(z*) λ - ν a + ν b = (Balance Forces) a z* b, c (z*) = (Feasibility) z*- a perp ν a b - z* perp ν b In terms of partitioned variables (when nown) Basic (n b = m): Β L (z*, λ, ν) = Β f(z*) + Β c(z*) λ = c (z*) = Square system in z B and λ Nonbasic (n n ): Ν L (z*, λ, ν) = Ν f(z*) + Ν c(z*) λ ν bnd = z*= bnd Square system in z N and ν Superbasic (n s ): S L (z*, λ, ν) = S f(z*) + S c(z*) λ = Square system in z S n = n n + n b + n s 1

11 Handling Basic Variables Β L (z*, λ, ν) = Β f(z*) + Β c(z*) λ = c (z*) = Full space: linearization and simultaneous solution of c(z) = with stationarity conditions. embedded within Newton method for KKT conditions (SQP and Barrier) Reduced space (two flavors): - elimination of nonlinear equations and z B at every point (feasible path GRG methods) - linearization, then elimination of linear equations and z B (MINOS and rsqp) 11

12 Handling Nonbasic Variables Ν L (z*, λ, ν) = Ν f(z*) + Ν c(z*) λ ν bnd = z*= bnd Barrier approach - primal and primal-dual approaches (IPOPT, KNITRO, LOQO) Active set (two flavors) QP or LP subproblems - primal-dual pivoting (SQP) gradient projection in primal space with bound constraints (GRG) 12

13 Gradient Projection Method (nonbasic variable treatment) Define the projection of an arbitrary point x onto box feasible region. The ith component is given by Piecewise linear path x(t) starting at the reference point x and obtained by projecting steepest descent direction at x onto the box region is given by 13

14 Handling Superbasic Variables S L (z*, λ, ν) = S f(z*) + S c(z*) λ = Second derivatives or quasi-newton? +1 B = B + T yy T s y - B s s T B s B s BFGS Formula s = x +1 -x y = L(x +1, u +1, v +1 ) - L(x, u +1, v +1 ) second derivatives approximated by change in gradients positive definite B ensures unique search direction and descent property Exact Hessian requires far fewer iterations (no buildup needed in space of superbasics) full-space - exploit sparsity (efficient large-scale solvers needed) reduced space - update or project (expensive dense algebra if ns large) 14

15 Algorithms for Constrained Problems Motivation: Build on unconstrained methods for superbasics wherever possible. Classification of Methods: Reduced Gradient Methods - (with Restoration) GRG2, CONOPT Reduced Gradient Methods - (without Restoration) MINOS Successive Quadratic Programming - generic implementations Barrier Functions - popular in 197s, but fell into disfavor. Barrier Methods have been developed recently and are again popular. Successive Linear Programming - only useful for "mostly linear" problems, cannot handle superbasic variables in a systematic way We will concentrate on algorithms for first four classes. 15

16 Reduced Gradient Methods Nonlinear elimination of basic variables (and multipliers, λ) Use gradient projection to update and remove nonbasic variables Solve problem in reduced space of superbasic variables using the concept of reduced gradients df dz dc df dz S S = f z f z S Because c( S z ) T c = z S dz S z B S c f z B [ c] z T f z B B [ c] dz B c c 1 = = z c S zb dz S z S z B This leads to reduced gradient for superbasic : = dz + dz =, we have : S c + zb 1 B 1 dz = 16

17 Example of Reduced Gradient Min s. t. c T x 2 1 3x 1 = [3 2 x x2 = 24 T 4], f = [2 x 1-2] Let df dz df dx S 1 z S = = f z = 2 x 1 x S 1, z 3 B z = S c x 2 [ c] z B 1 f z 1 [] 4 (- 2) = 2 x + 3 / 2 1 B If c T is (m x n); z S c T is m x (n-m); z B c T is (m x m) (df/dz S ) is the change in f along constraint direction per unit change in z S 17

18 Setch of GRG Algorithm 1. Initialize problem and obtain a feasible point at z 2. At feasible point z, partition variables z into z N, z B, z S 3. Remove nonbasics 4. Calculate reduced gradient, (df/dz S ) 5. Evaluate search direction for z S, d = B -1 (df/dz S ) 6. Perform a line search. Find α (,1] with z S := z S + α d Solve for c(z S + α d, z B, z N ) = If f(z S + α d, z B, z N ) < f(z S, z B, z N ), set z +1 S =z S + α d, := If (df/dz S ) <ε, Stop. Else, go to 2. 18

19 GRG Algorithm Properties 1. GRG2 has been implemented on PC's as GINO and is very reliable and robust. It is also the optimization solver in MS EXCEL. 2. CONOPT is implemented in GAMS, AIMMS and AMPL 3. GRG2 uses Q-N for small problems but can switch to conjugate gradients if problem gets large. 4. CONOPT does the same but reverts to an SQP method where it then uses exact second derivatives. This is a local feature. 5. Convergence of c(z S, z B, z N ) = can get very expensive because inner equation solutions are required. 6. Safeguards can be added so that restoration (step 5.) can be dropped and efficiency increases. 7. Line search used based on improvement of f(z). No global convergence proof 19

20 Reduced Gradient Method without Restoration Linearize then Eliminate Basics (MINOS/Augmented) Motivation: Efficient algorithms are available that solve linearly constrained optimization problems (MINOS): Min f(x) s.t. Ax b Cx = d Extend to nonlinear problems, through successive linearization Develop major iterations (linearizations) and minor iterations (GRG solutions). Strategy: (Robinson, Murtagh & Saunders) 1. Partition variables into basic, nonbasic variables and superbasic variables.. 2. Linearize active constraints at z D z= r 3. Let ψ = f (z) + v T c(z) + β (c(z) T c(z)) (Augmented Lagrange), 4. Solve linearly constrained problem: Min s.t. ψ (z) Dz = r a z b using reduced gradients to get z Set =+1, go to Algorithm terminates when no movement between steps 3) and 4). 2

21 MINOS/Augmented Notes 1. MINOS has been implemented very efficiently to tae care of linearity. It becomes LP Simplex method if problem is totally linear. Also, very efficient matrix routines. 2. No restoration taes place, nonlinear constraints are reflected in ψ(z) during step 3). MINOS is more efficient than GRG. 3. Major iterations (steps 3) - 4)) converge at a quadratic rate. 4. Reduced gradient methods are complicated, monolithic codes: hard to integrate efficiently into modeling software. 5. No globalization and no global convergence proof 21

22 Successive Quadratic Programming (SQP) Motivation: Tae KKT conditions, expand in Taylor series about current point. Tae Newton step (QP) to determine next point. Derivation KKT Conditions x L (x*, u*, v*) = f(x*) + ga(x*) u* + h(x*) v* = h(x*) = ga(x*) =, where g A are the active constraints. Newton -Step xx L g A h g T A h T Δx Δu = - Δv x L ( x, u, v ) g A h ( x ) ( x ) Requirements: xx L must be calculated and should be regular correct active set g A, need to now nonbasic variables! good estimates of u, v 22

23 SQP Chronology 1. Wilson (1963) - active set (nonbasics) can be determined by solving QP: Min f(x ) T d+ 1/2 d T xx L(x, u, v ) d d s.t. g(x ) + g(x ) T d h(x ) + h(x ) T d = 2. Han (1976), (1977), Powell (1977), (1978) - approximate xx L using a positive definite quasi-newton update (BFGS) - use a line search to converge from poor starting points. Notes: - Similar methods were derived using penalty (not Lagrange) functions. - Method converges quicly; very few function evaluations. - Not well suited to large problems (full space update used). For n > 1, say, use reduced space methods (e.g. MINOS). 23

24 Elements of SQP Search Directions How do we obtain search directions? Form QP and let QP determine constraint activity At each iteration,, solve: Min f(x ) T d + 1/2 d T B d d s.t. g(x ) + g(x ) T d h(x ) + h(x ) T d = Convergence from poor starting points As with Newton's method, choose α (stepsize) to ensure progress toward optimum: x +1 = x + α d. α is chosen by maing sure a merit function is decreased at each iteration. Exact Penalty Function ψ(x) = f(x) + μ [Σ max (, g j (x)) + Σ h j (x) ] μ > max j { u j, v j } Augmented Lagrange Function ψ(x) = f(x) + u T g(x) + v T h(x) + η/2 {Σ (h j (x)) 2 + Σ max (, g j (x)) 2 } 24

25 Newton-Lie Properties for SQP Fast Local Convergence B = xxl xxl is p.d and B is Q-N B is Q-N update, xxl not p.d Quadratic 1 step Superlinear 2 step Superlinear Enforce Global Convergence Ensure decrease of merit function by taing α 1 Trust region adaptations provide a stronger guarantee of global convergence - but harder to implement. 25

26 Basic SQP Algorithm. Guess x, Set B = I (Identity). Evaluate f(x ), g(x ) and h(x ). 1. At x, evaluate f(x ), g(x ), h(x ). 2. If >, update B using the BFGS Formula. 3. Solve: Min d f(x ) T d + 1/2 d T B d s.t. g(x ) + g(x ) T d h(x ) + h(x ) T d= If KKT error less than tolerance: L(x*) ε, h(x*) ε, g(x*) + ε. STOP, else go to Find α so that < α 1 and ψ(x + αd) < ψ(x ) sufficiently (Each trial requires evaluation of f(x), g(x) and h(x)). 5. x +1 = x + α d. Set = + 1 Go to 2. 26

27 Large-Scale SQP Min f(z) Min f(z ) T d + 1/2 d T W d s.t. c(z)= s.t. c(z ) + (Α ) T d = z L z z U z L z + d z U Active set strategy (usually) applied to QP problems (could use interior point) to handle nonbasic variables Few superbasics (1-1) Apply reduced space (linearized) elimination of basic variables Many superbasics ( 1) Apply full-space method 27

28 Few degrees of freedom => reduced space SQP (rsqp) Tae advantage of sparsity of A= c(x) project W into space of active (or equality constraints) curvature (second derivative) information only needed in space of degrees of freedom second derivatives can be applied or approximated with positive curvature (e.g., BFGS) use dual space QP solvers + easy to implement with existing sparse solvers, QP methods and line search techniques + exploits 'natural assignment' of dependent and decision variables (some decomposition steps are 'free') + does not require second derivatives - reduced space matrices are dense - may be dependent on variable partitioning - can be very expensive for many degrees of freedom - can be expensive if many QP bounds 28

29 29 Reduced space SQP (rsqp) Range and Null Space Decomposition = + ) ( ) ( T x c x f d A A W λ Assume nonbasics removed, QP problem with n variables and m constraints becomes: Define reduced space basis, Z R n x (n-m) with (A ) T Z = Define basis for remaining space Y R nx m, [Y Z ] R nx n is nonsingular. Let d = Y d Y + Z d Z to rewrite: [ ] [ ] [ ] = + ) ( ) ( T Z Y T T x c x f I Z Y d d I Z Y A A W I Z Y λ

30 3 Reduced space SQP (rsqp) Range and Null Space Decomposition (A T Y) d Y =-c(x ) is square, d Y determined from bottom row. Cancel Y T WY and Y T WZ; (unimportant as d Z, d Y --> ) (Y T A) λ = -Y T f(x ), λ can be determined by first order estimate Calculate or approximate w= Z T WY d Y, solve Z T WZ d Z =-Z T f(x )-w Compute total step: d = Y dy + Z dz = + ) ( ) ( ) ( T T Z Y T T T T T T x c x f Z x f Y d d Y A Z W Z Y W Z A Y Y W Y Y W Y λ

31 Barrier Methods for Large-Scale Nonlinear Programming Original Formulation min x R s.t n f ( x) c( x) = x Can generalize for a x b Barrier Approach min x R s.t n ϕ μ ( x) c( x) = = f ( x) μ ln n i= 1 x i As μ, x*(μ) x* Fiacco and McCormic (1968) 31

32 32 Solution of the Barrier Problem Newton Directions (KKT System) ) ( ) ( ) ( = = = + x c e Xv v x A x f μ λ Solve + = e Xv c v A f d d d X V A I A W x μ λ ν λ Reducing the System v Vd x X v e X d 1 1 =μ = Σ + + c d A A W x T ϕ μ λ V X 1 Σ= IPOPT Code IPOPT Code or.org ) ( [1,1,1...] x diag X e T = =

33 Global Convergence of Newton-based Barrier Solvers Merit Function Exact Penalty: P(x, η) = f(x) + η c(x) Aug d Lagrangian: L*(x, λ, η) = f(x) + λ T c(x) + η c(x) 2 Assess Search Direction (e.g., from IPOPT) Line Search choose stepsize α to give sufficient decrease of merit function using a step to the boundary rule with τ ~.99. for α (, α ], x v x + 1 = λ + α d + 1 = λ + α ( λ λ ) How do we balance φ (x) and c(x) with η? v x + α d Is this approach globally convergent? Will it still be fast? v + 1 = x (1 τ ) x (1 τ ) v + + α d > > x 33

34 Line Search Filter Method Store (φ, θ) at allowed iterates Allow progress if trial point is acceptable to filter with θ margin If switching condition T a b α[ φ d] δ[ θ ], a > 2b > is satisfied, only an Armijo line search is required on φ 2 φ(x) If insufficient progress on stepsize, evoe restoration phase to reduce θ. Global convergence and superlinear local convergence proved (with second order correction) θ(x) = c(x) 34

35 IPOPT Algorithm Features Line Search Strategies for Globalization - l 2 exact penalty merit function - Filter method (adapted and extended from Fletcher and Leyffer) Hessian Calculation - BFGS (full/lm) - SR1 (full/lm) - Exact full Hessian (direct) - Exact reduced Hessian (direct) - Preconditioned CG Algorithmic Properties Globally, superlinearly convergent (Wächter and B., 25) Easily tailored to different problem structures Freely Available CPL License and COIN-OR distribution: Beta version recently rewritten in C++ Solved on thousands of test problems and applications 35

36 Trust Region Barrier Method KNITRO W A A T d λ g = c 36

37 Composite Step Trust-region SQP Split barrier problem step into 2 parts: normal (basic) and tangential (nonbasic and superbasic) search directions Trust-region on tangential component: larger combined radius Zd θ δ 2 d d N δ 1 x 37

38 KNITRO Algorithm Features Overall method - Similar to IPOPT - But does not use filter method - applies full-space Newton step and line search (LOQO is similar) Trust Region Strategies for Globalization - based on l 2 exact penalty merit function - expensive: used only when line search approach gets into trouble Tangential Problem - Exact full Hessian (direct) Algorithmic Properties Globally, superlinearly convergent (Byrd and Nocedal, 2) Stronger convergence properties using trust region over line search Available through Zenia Available free to academics Solved on thousands of test problems and applications - Preconditioned CG to solve tangential problem Normal Problem - use dogleg method based on Cauchy step 38

39 Typical NLP algorithms and software SQP - NPSOL, VF2AD, fmincon reduced SQP - SNOPT, rsqp, MUSCOD, DMO, LSSOL GP + Elimination - GP/Lin. Elimin. - GRG2, GINO, SOLVER, CONOPT MINOS Second derivatives and barrier - IPOPT, KNITRO, LOQO Interesting hybrids - FSQP/cFSQP - SQP and elimination LANCELOT (actually AL for equalities along with GP) 39

40 Comparison of NLP Solvers: Data Reconciliation (24) Iterations Degrees of Freedom LANCELOT MINOS SNOPT KNITRO LOQO IPOPT 1 CPU Time (s, norm.) LANCELOT MINOS SNOPT KNITRO LOQO IPOPT.1 Degrees of Freedom 4

41 ari fail fail ex8_2_ i i ex8_2_ i i qcqp5-2c $ qcqp5-3c qcqp75-1c Mittelmann NLP benchmar ( ) qcqp75-2c t t tt qcqp1-1c 1 1c qcqp1-2c qcqp15-1c 1c t t t t t t fail fail tt qcqp5-2nc qcqp5-3nc IPOPT.6 qcqp75-1nc fail fail KNITRO qcqp75-2nc.5 LOQO t t tt SNOPT qcqp1-1nc.4 1nc CONOPT qcqp1-2nc qcqp15-1nc 1nc t t t t fail fail tt nql18 nql fail fail t t fail fail t t $ qssp6 qssp t t t t fail fail qssp m t t fail fail log(2)*minimum CPU time WM_CFx ! 3552! fail fail WM_CFy t t 462! 462! fail fail fail fail Weyl_m Large-scale t t 2772! 2772! fail fail Test Problems fail fail fail fail NARX_CFy ! 226! fail fail m fail fail tt 5-25 variables, 25 constraints NARX_Weyl t t 8291! 8291! fail fail fail fail fail fail fail fail =========================================================================== == fraction solved within Comparison of NLP solvers (latest 882 Mittelmann study) Limits Fail IPOPT 7 2 KNITRO 7 LOQO 23 4 SNOPT CONOPT

42 NLP Summary Widely used - solving nonlinear problems with thousands of variables and superbasics routinely Exploiting sparsity and structure leads to solving problems with millions of variables (and superbasics) Availability of structure and second derivatives is the ey. Global and fast local convergence properties are wellnown for many (but not all algorithms) Exceptions: LOQO, MINOS, CONOPT, GRG 42

43 Current Challenges - Exploiting larger problems in parallel (shared and distributed memory), - Irregular problems (e.g., singular problems, MPECs) - Extracting accurate first and second derivatives from codes... - Embedding robust NLPs within MINLP and global solvers in order to quicly detect infeasible solutions in subregions 43

MS&E 318 (CME 338) Large-Scale Numerical Optimization

Stanford University, Management Science & Engineering (and ICME) MS&E 318 (CME 338) Large-Scale Numerical Optimization 1 Origins Instructor: Michael Saunders Spring 2015 Notes 9: Augmented Lagrangian Methods