Nonlinear Programming: Concepts, Algorithms and Applications

Transcription

1 Nonlinear Programming: Concepts, Algorithms and Applications L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA Nonlinear Programming and Process Optimization Introduction Constrained Optimization Karush Kuhn-Tucer Conditions Reduced Gradient Methods (GRG, CONOPT, MINOS) Successive Quadratic Programming (SQP) Interior Point Methods (IPOPT) Process Optimization Blac Bo Optimization Modular Flowsheet Optimization Infeasible Path The Role of Eact Derivatives Large-Scale Nonlinear Programming rsqp: Real-time Process Optimization IPOPT: Blending and Data Reconciliation Further Applications Sensitivity Analysis for NLP Solutions Summary and Conclusions

2 Introduction Optimization: given a system or process, find the best solution to this process within constraints. Objective Function: indicator of "goodness" of solution, e.g., cost, yield, profit, etc. Decision Variables: variables that influence process behavior and can be adjusted for optimization. In many cases, this tas is done by trial and error (through case study). Here, we are interested in a systematic approach to this tas - and to mae this tas as efficient as possible. Some related areas: - Math programming - Operations Research Currently - Over 3 journals devoted to optimization with roughly papers/month - a fast moving field! 3 Optimization Viewpoints Mathematician - characterization of theoretical properties of optimization, convergence, eistence, local convergence rates. Numerical Analyst - implementation of optimization method for efficient and "practical" use. Concerned with ease of computations, numerical stability, performance. Engineer - applies optimization method to real problems. Concerned with reliability, robustness, efficiency, diagnosis, and recovery from failure. 4

3 Motivation Scope of optimization Provide systematic framewor for searching among a specified space of alternatives to identify an optimal design, i.e., as a decision-maing tool Premise Conceptual formulation of optimal product and process design corresponds to a mathematical programming problem min R f (, y) st h(, y) = MINLP g(, y) n y {,} ny MINLP NLP 5 Optimization in Design, Operations and Control MILP MINLP Global LP,QP NLP SA/GA HENS MENS Separations Reactors Equipment Design Flowsheeting Scheduling Supply Chain Real-time optimization Linear MPC Nonlinear MPC Hybrid 6 3

4 Constrained Optimization (Nonlinear Programming) Problem: Min f() s.t. g() h() = where: f() - scalar objective function - n vector of variables g() - inequality constraints, m vector h() - meq equality constraints. Sufficient Condition for Global Optimum - f() must be conve, and - feasible region must be conve, i.e. g() are all conve h() are all linear Ecept in special cases, there is no guarantee that a local optimum is global if sufficient conditions are violated. 7 Eample: Minimize Pacing Dimensions What is the smallest bo for three round objects? Variables: A, B, (, y ), (, y ), ( 3, y 3 ) Fied Parameters: R, R, R 3 Objective: Minimize Perimeter = (A+B) Constraints: Circles remain in bo, can't overlap Decisions: Sides of bo, centers of circles. A y 3, y R B - R, y A - R ( - ) + ( y - y ) ( R + R), y R B - R, y A - R ( - 3) + ( y - y 3) ( R + R3) 3, y 3 R 3 3 B - R 3, y 3 A - R 3 ( - 3) + ( y - y 3) ( R + R3) in bo,, 3, y, y, y 3, A, B B no overlaps 8 4

5 Characterization of Constrained Optima Min Min Linear Program Linear Program (Alternate Optima) Min Min Min Conve Objective Functions Linear Constraints Min Min Min Min Nonconve Region Multiple Optima Min Nonconve Objective Multiple Optima 9 What conditions characterize an optimal solution? Unconstrained Local Minimum Necessary Conditions f (*) = p T f (*) p for p R n (positive semi-definite) Unconstrained Local Minimum Sufficient Conditions f (*) = p T f (*) p > for p R n (positive definite) 5

6 Optimal solution for inequality constrained problem Min f() s.t. g() Analogy: Ball rolling down valley pinned by fence Note: Balance of forces ( f, g) Optimal solution for general constrained problem Problem: Min f() s.t. g() h() = Analogy: Ball rolling on rail pinned by fences Balance of forces: f, g, h 6

7 Optimality conditions for local optimum Necessary First Order Karush Kuhn - Tucer Conditions L (*, u, v) = f(*) + g(*) u + h(*) v = (Balance of Forces) u (Inequalities act in only one direction) g (*), h (*) = (Feasibility) u j g j (*) = (Complementarity: either g j (*) = 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 (*) h(*)], to be linearly independent. Also, under LICQ, KKT multipliers are uniquely determined. Necessary (Sufficient) Second Order Conditions - Positive curvature in "constraint" directions. - p T L (*) p (p T L (*) p > ) where p are the constrained directions: g A (*) T p =, h(*) T p = 3 Single Variable Eample of KKT Conditions Min () s.t. -a a, a > * = is seen by inspection f() Lagrange function : L(, u) = + u (-a) + u (-a-) First Order KKT conditions: L(, u) = + u -u = u (-a) = u (-a-) = -a a u, u -a a Consider three cases: u, u = Upper bound is active, = a, u = -a, u = u =, u Lower bound is active, = -a, u = -a, u = u = u = Neither bound is active, u =, u =, = Second order conditions (*, u, u =) L (*, u*) = p T L (*, u*) p = (Δ) > 4 7

8 Min -() s.t. -a a, a > * = ±a is seen by inspection Single Variable Eample of KKT Conditions - Revisited -a a Lagrange function : L(, u) = - +u( (-a) + u (-a-) First Order KKT conditions: L(, u) = - + u -u = u (-a) = u (-a-) = -a a u, u f() Consider three cases: u, u = Upper bound is active, = a, u = a, u = u =, u Lower bound is active, = -a, u = a, u = u = u = Neither bound is active, u =, u =, = Second order conditions (*, u, u =) L (*, u*) = - p T L (*, u*) p = -(Δ) < 5 Interpretation of Second Order Conditions For = a or = -a, we require the allowable direction to satisfy the active constraints eactly. Here, any point along the allowable direction, * must remain at its bound. For this problem, however, there are no nonzero allowable directions that satisfy this condition. Consequently the solution * is defined entirely by the active constraint. The condition: p T L (*, u*, v*) p > for the allowable directions, is vacuously satisfied - because there are no allowable directions that satisfy g A (*) T p =. Hence, sufficient second order conditions are satisfied. As we will see, sufficient second order conditions are satisfied by linear programs as well. 6 8

9 Role of KKT Multipliers -a a a + Δa Also nown as: Shadow Prices Dual Variables Lagrange Multipliers f() Suppose a in the constraint is increased to a + Δa f(*) = (a + Δa) and [f(*, a + Δa) - f(*, a)]/δa = a + Δa df(*)/da = a = u 7 Another Eample: Constraint Qualifications Min s. t. 3 ( ) * = * = 3( ) u + u -, u, u = ( ) 3, u, u ( 3 ( ) ) = KKT conditions not satisfied at NLP solution Because a CQ is not satisfied (e.g., LICQ) 8 9

10 Algorithms for Constrained Problems Motivation: Build on unconstrained methods wherever possible. Classification of Methods: Reduced Gradient Methods - (with Restoration) GRG, CONOPT Reduced Gradient Methods - (without Restoration) MINOS Successive Quadratic Programming - generic implementations Penalty Functions - popular in 97s, but fell into disfavor. Barrier Methods have been developed recently and are again popular. Successive Linear Programming - only useful for "mostly linear" problems We will concentrate on algorithms for first four classes. Evaluation: Compare performance on "typical problem," cite eperience on process problems. 9 Reduced Gradient Method with Restoration (GRG/CONOPT) Min f() Min f(z) s.t. g() + s = (add slac variable) ` s.t. c(z) = h() = a z b a b, s Partition variables into: z B - dependent or basic variables z N - nonbasic variables, fied at a bound z S - independent or superbasic variables Modified KKT Conditions f (z) + c(z)λ ν L + ν U = c(z) = z (i) (i) = z U or z (i) = z (i) L, i N ν U (i), ν L (i) =, i N

11 Reduced Gradient Method with Restoration (GRG/CONOPT) a) S f (z) + S c(z)λ = b) B f (z) + B c(z)λ = c) N f (z) + N c(z)λ ν L + ν U = d) z (i) (i) = z U or z (i) = z (i) L, i N e) c(z) = z B = z B (z S ) Solve bound constrained problem in space of superbasic variables (apply gradient projection algorithm) Solve (e) to eliminate z B Use (a) and (b) to calculate reduced gradient wrt z S. Nonbasic variables z N (temporarily) fied (d) Repartition based on signs of ν, if z s remain at bounds or if z B violate bounds Definition of Reduced Gradient df = f + dz B f dz S z S dz S z B Because c(z) =,we have : T dc = c dz S + c dz B = z S z B dz B = c c dz S z S z B This leads to : df = S f (z) S c B c dz S T = zs c[ zb c] [ ] B f (z) = S f (z) + S c(z)λ By remaining feasible always, c(z) =, a z b, one can apply an unconstrained algorithm (quasi-newton) using (df/dz S ), using (b) Solve problem in reduced space of z S variables, using (e).

12 Eample of Reduced Gradient Min s. t. c T = [3 4], f = 4 T = [ - ] Let df dz df d S z S =, z f = z S B = z S [ c] [ 4 ] ( - ) = + 3 / = 3 c z B f z B If c T is (m n); z S c T is m (n-m); z B c T is (m m) (df/dz S ) is the change in f along constraint direction per unit change in z S 3 Gradient Projection Method (superbasic nonbasic variable partition) Define the projection of an arbitrary point onto bo feasible region. The ith component is given by Piecewise linear path (t) starting at the reference point and obtained by projecting steepest descent (or any search) direction at onto the bo region is given by where g is the reduced gradient, t is the stepsize. Also, can adapt to (quasi-) Newton method. 4

13 Setch of GRG Algorithm. Initialize problem and obtain a feasible point at z. At feasible point z, partition variables z into z N, z B, z S 3. Cl Calculate lt reduced dgradient, t(df/d (df/dz S ) 4. Evaluate search direction for z S, d = B - (df/dz S ) 5. Perform a line search. Find α (,] with z S := z S + α d Solve for c(z S + α d, z B B, z N N) = If f(z S + α d, z B, z N ) < f(z S, z B, z N ), set z + S =z S + α d, := + 6. If (df/dz S ) <ε, Stop. Else, go to. 5 Reduced Gradient Method with Restoration z B z S 6 3

14 Reduced Gradient Method with Restoration z B Fails, due to singularity in basis matri (dc/dz B ) z S 7 Reduced Gradient Method with Restoration z B Possible remedy: repartition basic and superbasic variables to create nonsingular basis matri (dc/dz B ) z S 8 4

15 GRG Algorithm Properties. GRG has been implemented on PC's as GINO and is very reliable and robust. It is also the optimization solver in MS EXCEL.. CONOPT is implemented in GAMS, AIMMS and AMPL 3. GRG uses Q-N for small problems but can switch to conjugate gradients if problem gets large. CONOPT uses eact second derivatives. 4. Convergence of c(z S, z B, z N ) = can get very epensive because c(z) is calculated repeatedly. 5. Safeguards can be added so that restoration (step 5.) can be dropped and efficiency increases. Representative Constrained Problem Starting Point [.8,.] GINO Results - 4 iterations to f(*) -6 CONOPT Results - 7 iterations to f(*) -6 from feasible point. 9 Reduced Gradient Method without Restoration z B z S 3 5

16 Reduced Gradient Method without Restoration (MINOS/Augmented) Motivation: Efficient algorithms are available that solve linearly constrained optimization i problems (MINOS): Min f() s.t. A b C = d Etend to nonlinear problems, through successive linearization Develop major iterations (linearizations) and minor iterations (GRG solutions). Strategy: (Robinson, Murtagh & Saunders). Partition variables into basic, nonbasic variables ibl and superbasic variables.. ibl. Linearize active constraints at z D z = r 3. Let ψ = f (z) + λ T c (z) + β (c(z) T c(z)) (Augmented Lagrange), 4. Solve linearly constrained problem: Min ψ (z) s.t. Dz = r a z b using reduced gradients to get z + 5. Set =+, go to. 6. Algorithm terminates when no movement between steps ) and 4). 3 MINOS/Augmented Notes. MINOS has been implemented very efficiently to tae care of linearity. It becomes LP Simple method if problem is totally linear. Also, very efficient matri routines.. 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. Representative Constrained Problem Starting Point [.8,.] MINOS Results: 4 major iterations, function calls to f(*)

17 Successive Quadratic Programming (SQP) Motivation: Tae KKT conditions, epand in Taylor series about current point. Tae Newton step (QP) to determine net point. Derivation KKT Conditions L (*, u*, v*) = f(*) + ga(*) u* + h(*) v* = h(*) = ga(*) =, where g A are the active constraints. Newton -Step L g A h g T A h T Δ Δu = - Δv Requirements: L must be calculated and should be regular correct active set g A good estimates of u, v L (, u, v ) g A ( ) h ( ) 33 SQP Chronology. Wilson (963) - active set can be determined by solving QP: Min f( ) T d + / d T L(, u, v ) d d s.t. g( ) + g( ) T d h( ) + h( ) T d =. Han (976), (977), Powell (977), (978) - approimate 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 >, say, use reduced space methods (e.g. MINOS). 34 7

18 Elements of SQP Hessian Approimation What about L? need to get second derivatives for f(), g(), h(). need to estimate multipliers, u, v ; L may not be positive semidefinite Approimate L (, u, v ) by B, a symmetric positive definite matri. BFGS Formula T + yy B = B + T s y - B s T s B s B s s = + - y = L( +, u +, v + ) - L(, u +, v + ) second derivatives approimated by change in gradients positive definite B ensures unique QP solution Newton-lie properties leads to superlinear convergence. 35 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( ) T d + / d T B d d s.t. g( ) + g( ) T d h( ) + h( ) T d = Convergence from poor starting points As with Newton's method, choose α (stepsize) to ensure progress toward optimum: + = + α d. α is chosen by maing sure a merit function is decreased at each iteration. Eact Penalty Function ψ() = f() + μ [Σ ma (, g j ()) + Σ h j () ] μ > ma j { u j, v j } Augmented Lagrange Function ψ() = f() + u T g() + v T h() + η/ {Σ (h j ()) + Σ ma (, g j ()) } 36 8

19 Basic SQP Algorithm. Guess, Set B = I (Identity). Evaluate f( ), g( ) and h( ).. At, evaluate f( ), g( ), h( ).. If >, update B using the BFGS Formula. 3. Solve: Min d f( ) T d + / d T B d s.t. g( ) + g( ) T d h( ) + h( ) T d = If KKT error less than tolerance: L(*) ε, h(*) ε, g(*) + ε. STOP, else go to Find α so that < α and ψ( + αd) < ψ( ) sufficiently (Each trial requires evaluation of f(), g() and h()) = + α d. Set = + Go to. 37 SQP Test Problem * Min s.t (- ) - (- ) 3 * = [.5,.375]. 38 9

20 SQP Test Problem First Iteration Start from the origin ( = [, ] T ) with B = I, form: Min d + / (d + d ) s.t. d d + d d = [, ] T. with μ = and μ =. 39 SQP Test Problem Second Iteration * From = T [.5, ] with B = I (no update from BFGS possible), form:. Min d + / (d + d ) s.t. -.5 d -d d -d d = [,.375] T with μ =.5 and μ =.5 * = [.5,.375] T is optimal 4

21 Problems with SQP Nonsmooth Functions - Reformulate Ill-conditioning - Proper scaling Poor Starting Points Trust Regions can help Inconsistent Constraint Linearizations - Can lead to infeasible QP s - Analogy to Singular Jacobian in Newton s Method Min s.t. + -( ) - -( ) -/ 4 Simulation and Optimization of Flowsheets w(y) h(y) = y 5 Original Problem Transformed Problem Min f() Min f(, y) s.t. g() s.t. g(, y) h(, y) = y w(, y) = Characteristics replace recycle convergence with optimization bloc models and sequencing manager intact. larger optimization problem gradients evaluated from flowsheet passes avoids repeated (epensive) recycle convergence 4

22 Ammonia Process Optimization H N P r T r Hydrogen Feed Nitrogen Feed N 5.% 99.8% H 94.% --- CH4.79 %.% Ar ---.% ν Tf Tf T o Product Hydrogen and Nitrogen feed are mied, compressed, and combined with recycle & brought to reactor temperature. Reaction occurs in a multibed reactor (equilibrium reactor) Reactor effluent is cooled and product is separated with two flashes Liquid yields high purity NH 3 product. Vapor is recompressed and recycled. 43 Ammonia Process Optimization Optimization Problem Performance Characterstics Ma {Total 5% over five years} 5 SQP iterations.. base point simulations. s.t. 5 tons NH3/yr. objective function improves by Pressure Balance No Liquid in Compressors $.66 6 to $ H/N 3.5 difficult to converge flowsheet Treact o F NH3 purged 4.5 lb mol/hr at starting point NH3 Product Purity 99.9 % Item Optimum Starting point Tear Equations Objective Function($ 6 ) Inlet temp. reactor ( o F) 4 4. Inlet temp. st flash ( o F) Inlet temp. nd flash ( o F) Inlet temp. rec. comp. ( o F) Purge fraction (%) Reactor Press. (psia) Feed (lb mol/hr) Feed (lb mol/hr)

23 How accurate should gradients be for optimization? Recognizing True Solution KKT conditions and Reduced Gradients determine true solution Derivative Errors will lead to wrong solutions! Performance of Algorithms Constrained NLP algorithms are gradient based (SQP, Conopt, GRG, MINOS, etc.) Global and Superlinear convergence theory assumes accurate gradients Worst Case Eample (Carter, 99) Newton s Method generates an ascent direction and fails for any ε! Min T f ( ) = A ε + / ε ε / ε A = / / ε ε ε + ε = [ ] f ( ) = ε g( ) = f ( ) + O( ε ) T d = A g( ) κ ( A) = (/ ε ) f 45 Implementation of Analytic Derivatives parameters, p eit variables, s Module Equations c(v,, s, p, y) = y Automatic Differentiation Tools Sensitivity Equations dy/d ds/d dy/dp ds/dp JAKE-F, limited it to a subset of FORTRAN (Hillstrom, 98) DAPRE, which has been developed for use with the NAG library (Pryce, Davis, 987) ADOL-C, implemented using operator overloading features of C++ (Griewan, 99) ADIFOR, (Bischof et al, 99) uses source transformation approach FORTRAN code. TAPENADE, web-based source transformation for FORTRAN code Relative effort needed to calculate gradients is not n+ but about 3 to 5 (Wolfe, Griewan) 46 3

24 Flash Recycle Optimization ( decisions + 7 tear variables) Ammonia Process Optimization (9 decisions and 6 tear variables) S3 S Mier S Flash P S6 S5 S4 Ratio 3 / Ma S3(A) *S3(B) - S3(A) - S3(C) + S3(D) - (S3(E)) S7 8 Lo P GRG SQP GRG Flash SQP 6 rsqp CPU Seconds (VS 3) rsqp CPU Seconds (VS 3) 4 Numerical Eact Numerical Eact 47 Large-Scale SQP Min f(z) Min f(z ) T d + / 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 ( - ) Apply reduced space (linearized) elimination of basic variables Many superbasics ( ) Apply full-space method 48 4

25 Few degrees of freedom => reduced space SQP (rsqp) Tae advantage of sparsity of A= c() 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 approimated with positive curvature (e.g., BFGS) use dual space QP solvers + easy to implement with eisting sparse solvers, QP methods and line search techniques + eploits '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 epensive for many degrees of freedom - can be epensive if many QP bounds 49 Reduced space SQP (rsqp) Range and Null Space Decomposition Assume nonbasics removed, QP problem with n variables and m constraints becomes: W A d f ( ) = A T λ+ c( ) Define reduced space basis, Z R n (n-m) with (A ) T Z = Define basis for remaining space Y R n m, [Y Z ] R n n is nonsingular. Let d = Y d Y + Z d Z to rewrite: T d [ ] Y [ ] Y Z W A Y Z [ Y Z ] I A T = dz I λ + T f ( ) I c( ) 5 5

26 Reduced space SQP (rsqp) Range and Null Space Decomposition Y T W Y T Z W Y T A Y T Y W Y Y Z T W Z A d Y Y d = Z Z λ T T T f ( f ( c( + ) ) ) (A T Y) d Y =-c( ) is square, d Y determined from bottom row. Cancel lyy T WY and Y T WZ; (unimportant tas d Z, d Y --> >) (Y T A) λ = -Y T f( ), λ can be determined by first order estimate Calculate or approimate w= Z T WY d Y, solve Z T WZ d Z =-Z T f( )-w Compute total step: d = Y dy + Z dz 5 Reduced space SQP (rsqp) Interpretation d Z d Y Range and Null Space Decomposition SQP step (d) operates in a higher dimension Satisfy constraints using range space to get d Y Solve small QP in null space to get d Z In general, same convergence properties as SQP. 5 6

27 Choice of Decomposition Bases. Apply QR factorization to A. Leads to dense but well-conditioned Y and Z. R A = Q = R [ Y Z ]. Partition variables into decisions u and dependents v. Create orthogonal Y and Z with embedded identity matrices (A T Z =, Y T Z=). A T = I Z = C T T [ c c ] = [ N C] u N T N C Y = I 3C 3. Coordinate Basis - same Z as above, Y T = [ I] v Bases use gradient information already calculated. Adapt decomposition to QP step Theoretically same rate of convergence as original SQP. Coordinate basis can be sensitive to choice of u and v. Orthogonal is not. Need consistent initial point and nonsingular C; automatic generation T 53 rsqp Algorithm. Choose starting point.. At iteration, evaluate functions f( ), c( ) and their gradients. 3. Calculate bases Y and Z. 4. Solve for step d Y in Range space from (A T Y) d Y =-c( ) 5. Update projected Hessian B ~ Z T WZ (e.g. with BFGS), w (e.g., zero) 6. Solve small QP for step d Z in Null space. Min s. t. T T ( Z f ( ) + w ) d L + Yd Y + Zd Z Z + / d U T Z B d 7. If error is less than tolerance stop. Else 8. Solve for multipliers using (Y T A) λ = -Y T f( ) 9. Calculate total step d = Y dy + Z dz.. Find step size α and calculate new point, + = + α d 3. Continue from step with = +. Z 54 7

28 Comparison of SQP and rsqp Coupled Distillation Eample - 5 Equations Decision Variables - boilup rate, reflu ratio Method CPU Time Annual Savings Comments. SQP* hr negligible Base Case. rsqp 5 min. $ 4, Base Case 3. rsqp 5 min. $ 84, Higher Feed Tray Location 4. rsqp 5 min. $ 84, Column Overhead to Storage 5. rsqp 5 min $7, Cases 3 and 4 together 8 Q VK Q VK 55 Real-time Optimization: State of Art w APC Plant u RTO c(, u, p) = On line optimization Steady state model for states () Supply setpoints (u) to APC (control system) Model mismatch, measured and unmeasured disturbances (w) Min u F(, u, w) s.t. c(, u, p, w) = X, u U p y DR-PE c(, u, p) = Data Reconciliation & Parameter Identification Estimation problem formulations Steady state model Maimum lielihood objective functions considered to get parameters (p) Min p Φ(, y, p, w) s.t. c(, u, p, w) = X, p P 56 8

29 Real-time Optimization with rsqp Sunoco Hydrocracer Fractionation Plant (Bailey et al, 993) Eisting process, optimization on-line at regular intervals: 7 hydrocarbon components, 8 heat echangers, absorber/stripper (3 trays), debutanizer ( trays), C3/C4 splitter ( trays) and deisobutanizer (33 trays). REACTOR EFFLUENT FROM LOW PRESSURE SEPARATOR FUEL GAS ABSORBER/ STRIPPER C3 PREFLASH LIGHT NAPHTHA MAIN FRAC. REFORMER NAPHTHA MIXED LPG DEBUTANIZER C3/C4 SPLITTER DIB ic4 RECYCLE OIL nc4 square parameter case to fit the model to operating data. optimization to determine best operating conditions 57 Optimization Case Study Characteristics Model consists of 836 equality constraints and only ten independent variables. It is also reasonably sparse and contains 43 nonzero Jacobian elements. NP G E P P = z i C i + z i C i + z i C m i U i G i E m= Cases Considered:. Normal Base Case Operation. Simulate fouling by reducing the heat echange coefficients for the debutanizer 3Si 3. Simulate fouling by reducing the heat echange coefficients ffii for splitter feed/bottoms echangers 4. Increase price for propane 5. Increase base price for gasoline together with an increase in the octane credit 58 9

30 59 Many degrees of freedom => full space IPOPT W + Σ A T A d ϕ( = λ c( + ) ) wor in full space of all variables second derivatives useful for objective and constraints use specialized large-scale Newton solver + W= L(,λ) and A= c() sparse, often structured + fast if many degrees of freedom present + no variable partitioning required - second derivatives strongly desired - W is indefinite, requires comple stabilization - requires specialized large-scale linear algebra 6 3

31 Barrier Methods for Large-Scale Nonlinear Programming min f ( ) Original Formulation R s.t n c ( ) = Can generalize for a b Barrier Approach min n R ϕ μ s.t c( ) = ( ) = f ( ) μ ln i n i= As μ, *(μ) * Fiacco and McCormic (968) 6 Solution of the Barrier Problem e Newton Directions (KKT System) T f = [,,...], X = diag( ) A = c( ), W = ( ) + A( ) λ v L(, λ, ν ) Xv μeμ e c ( ) = = = Solve Reducing the System d =μx e v X v Vd W I d f + A λ v Σ A d T = A = ϕ μ d c + λ V X λ d c Xv μ e ν W + d T A Σ= X V IPOPT Code 6 3

32 Global Convergence of Newton-based Barrier Solvers Merit Function Eact Penalty: P(, η) = f() + η c() Aug d Lagrangian: L*(, λ, η) = f() + λ T c() + η c() 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 ], + = + α d ( τ ) v + + = v + α d λ + α d > ( τ ) v = λ + α ( λ λ ) How do we balance φ () and c() with η? Is this approach globally convergent? Will it still be fast? v + > 63 Global Convergence Failure (Wächter and B., ) s. t. ( ) Min f ( ) = 3 =, 3 Newton-type line search stalls even though descent directions eist T A ( ) d + c ( ) = + α d > Remedies: Composite Step Trust Region (Byrd et al.) Filter Line Search Methods 64 3

33 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 > b > is satisfied, only an Armijo line search is required on φ If insufficient progress on stepsize, evoe restoration phase to reduce θ. Global convergence and superlinear local convergence proved (with second order correction) φ() θ() = c() 65 Implementation Details Modify KKT (full space) matri if nonsingular W + Σ + δ A T A δ I δ - Correct inertia to guarantee descent direction δ - Deal with ran deficient A KKT matri factored by MA7 Feasibility restoration phase Min c ( ) + l Q Apply Eact Penalty Formulation Eploit same structure/algorithm to reduce infeasibility u 66 33

34 IPOPT Algorithm Features Line Search Strategies for Globalization - l eact penalty merit function - augmented Lagrangian merit function - Filter method (adapted and etended from Fletcher and Leyffer) Hessian Calculation - BFGS (full/lm and reduced space) - SR (full/lm and reduced space) - Eact full Hessian (direct) - Eact reduced Hessian (direct) - Preconditioned CG Algorithmic Properties Globally, superlinearly convergent (Wächter and B., 5) Easily tailored to different problem structures Freely Available CPL License and COIN-OR distribution: IPOPT 3. recently rewritten in C++ Solved on thousands of test problems and applications 67 Gasoline Blending OIL TANKS Gasoline Blending Here GAS STATIONS Supply tans Intermediate tans FINAL PRODUCT TRUCKS Final Product tans 68 34

35 Blending Problem & Model Formulation q t, i q Supply tans (i) i f t, ij.. q t, j..q t, f t, j v v t, j Intermediate tans (j).. ma c f c ) f t, Final Product tans () f & v flowrates and tan volumes q tan qualities ( f t, i, t i t i s.t. f f + v = v t, j t, ij t +, j t, j i f t, f = t, j j q f q f + q t, j t, j t, i t, ij t +, j i q f q f = t, t, t, j t, j j q q t, q ma min v j v t, j v j min ma v t = q v +, j t, j t, j Model Formulation in AMPL 69 Small Multi-day Blending Models Single Qualities Haverly, C. 978 (HM) Audet & Hansen 998 (AHM) F B F B F P F P B F3 B F3 P B3 7 35

36 Honeywell Blending Model Multiple Days 48 Qualities 7 Comparison of NLP Solvers: Data Reconciliation terations It Degrees of Freedom LANCELOT MINOS SNOPT KNITRO LOQO IPOPT CPU Time (s, norm m.) 4 6. LANCELOT MINOS SNOPT KNITRO LOQO IPOPT. Degrees of Freedom 7 36

37 Comparison of NLP solvers (latest Mittelmann study) Mittelmann NLP benchmar (-6-8) fraction solved within IPOPT KNITRO LOQO SNOPT CONOPT Limits Fail IPOPT 7 KNITRO 7 LOQO 3 4 SNOPT 56 CONOPT log()*minimum CPU time 7 Large-scale Test Problems 5-5 variables, 5 constraints 73 Typical NLP algorithms and software SQP - NPSOL, VFAD, NLPQL, fmincon reduced SQP - SNOPT, rsqp, MUSCOD, DMO, LSSOL Reduced Grad. rest. - GRG, GINO, SOLVER, CONOPT Reduced Grad no rest. - MINOS Second derivatives and barrier - IPOPT, KNITRO, LOQO Interesting hybrids - FSQP/cFSQP - SQP and constraint elimination LANCELOT (Augmented Lagrangian w/ Gradient Projection) 74 37

38 Rules for Formulating Nonlinear Programs ) Avoid overflows and undefined terms, (do not divide, tae logs, etc.) e.g. + y - ln z = + y - u = ep u - z = ) If constraints must always be enforced, mae sure they are linear or bounds. eg e.g. v(y - z ) / =3 vu = 3 u -(y-z ) =, u 3) Eploit linear constraints as much as possible, e.g. mass balance i L + y i V = F z i l i + v i = f i L l i = 4) Use bounds and constraints to enforce characteristic solutions. e.g. a b, g () to isolate correct root of h () =. 5) Eploit global properties p when possibility eists. Conve (linear equations?) Linear Program? Quadratic Program? Geometric Program? 6) Eploit problem structure when possible. e.g. Min [T - 3Ty] s.t. T + y - T y = 5 4-5Ty + T = 7 T (If T is fied solve LP) put T in outer optimization loop. 75 Sensitivity Analysis for Nonlinear Programming At nominal conditions, p Min f(, p ) s.t. c(, p ) = a(p ) b(p ) How is the optimum affected at other conditions, p p? Model parameters, prices, costs Variability in eternal conditions Model structure How sensitive is the optimum to parametric uncertainties? Can this be analyzed easily? 76 38

39 Second Order Optimality Conditions: Reduced Hessian needs to be positive semi-definite - Nonstrict local minimum: If nonnegative, find eigenvectors for zero eigenvalues, regions of nonunique solutions - Saddle point: If any are eigenvalues are negative, move along directions of corresponding eigenvectors and restart optimization. z Saddle Point * z 77 IPOPT Factorization Byproducts: Tools for Postoptimality and Uniqueness Modify KKT (full space) matri if nonsingular W + Σ + δi A A T δi δ - Correct inertia to guarantee descent direction δ - Deal with ran deficient A KKT matri factored by indefinite symmetric factorization Solution with δ, δ = sufficient second order conditions Eigenvalues of reduced Hessian all positive unique minimizer and multipliers Else: Reduced Hessian available through sensitivity calculations Find eigenvalues to determine nature of stationary point 78 39

40 NLP Sensitivity Parametric Programming Solution Triplet Optimality Conditions NLP Sensitivity Rely upon Eistence and Differentiability of Path Main Idea: Obtain and find by Taylor Series Epansion 79 NLP Sensitivity Properties (Fiacco, 983) Assume sufficient differentiability, LICQ, SSOC, SC: Intermediate IP solution (s(μ)-s*) = O(μ) Finite neighborhood around p and μ= with same active set eists and is unique 8 4

41 NLP Sensitivity Obtaining Optimality Conditions of Apply Implicit Function Theorem to around KKT Matri IPOPT Already Factored at Solution Sensitivity Calculation from Single Bacsolve Approimate Solution Retains Active Set 8 Sensitivity for Flash Recycle Optimization ( decisions, 7 tear variables) S3 S Mier S Flash P S6 S5 S4 Ratio 3 / Ma S3(A) *S3(B) - S3(A) - S3(C) + S3(D) - (S3(E)) S7 Second order sufficiency test: Dimension of reduced Hessian = Positive eigenvalue Sensitivity to simultaneous change in feed rate and upper bound on purge ratio 8 4

42 Ammonia Process Optimization (9 decisions, 8 tear variables) Objective Function Sensitivities vs. Re-optimized Pts Reactor Sensitivity QP QP Actual Second order sufficiency test: Dimension of reduced Hessian Lo = P4 Eigenvalues = [.8E-4, 8.3E-,.8E-4, 7.7E-5] Sensitivity to simultaneous change Flash in feed rate and upper bound on reactor conversion 7. Hi P Flash Relative perturbation change.. 83 Hierarchy of Nonlinear Programming Formulations and Model Intrusion Full Space Formulation (GAMS, AMPL ) Open Interior Point Compute Efficiency Real-time Optimization Eact Derivatives rsqp SQP Blac Bo DFO Closed 4 6 Variables/Constraints 84 4

43 Summary and Conclusions Optimization Algorithms -KKT Conditions -Reduced Gradient Methods (GRG, MINOS) -Successive Quadratic Programming (SQP) -Reduced Hessian SQP -Interior Point NLP (IPOPT) Process Optimization Applications -Modular Flowsheet Optimization -Equation Oriented Models and Optimization -Realtime Process Optimization -Blending with many degrees of freedom Further Applications -Sensitivity Analysis for NLP Solutions 85 43