A Branch-and-Refine Method for Nonconvex Mixed-Integer Optimization

Transcription

1 A Branch-and-Refine Method for Nonconvex Mixed-Integer Optimization Sven Leyffer 2 Annick Sartenaer 1 Emilie Wanufelle 1 1 University of Namur, Belgium 2 Argonne National Laboratory, USA IMA Workshop, Minneapolis, November 18, 2008

2 Overview Application & Motivation Tertiary Voltage Control Problem Characteristics Branch-and-Refine for Nonconvex MINLPs Special Ordered Set (SOS) Approximations Decomposition of Nonlinear Functions Piecewise Polyhedral Envelopes Branch-and-Refine Theoretical and Numerical Results Comparison to Underestimators & SOS-Branching Numerical Experience Conclusions & Future Research

3 Overview Application & Motivation Tertiary Voltage Control Problem Characteristics Branch-and-Refine for Nonconvex MINLPs Special Ordered Set (SOS) Approximations Decomposition of Nonlinear Functions Piecewise Polyhedral Envelopes Branch-and-Refine Theoretical and Numerical Results Comparison to Underestimators & SOS-Branching Numerical Experience Conclusions & Future Research

4 Tertiary Voltage Control Problem Provided by Tractebel Engineering Optimal Power Flow (OPF) problem: find the best operating conditions of an existing system Focus on the Tertiary Voltage Control (TVC) problem (but applicable to more general OPF problems) In alternating current, power is a complex number: real part = real power (P) imaginary part = reactive power (Q) Reactive power transmission causes voltage drops and losses need a regulation of the reactive power produced by each generator

5 TVC min k N G w k (Q k obj k ) 2, s.t. P i P ic P ik P ik P ik P ik = 0, i N, ik S s i ik S e i ik T s i Q i Q ic + a i ν 2 i Q i0 Q ik Q ik Q ik Q ik = 0, i N, ik S s i ik S e i ik T e i ik T s i ik T e i Q ik = K, ik B ν mini ν i ν maxi, a i binary, i N, P mini P i P maxi, i N G, Q mini Q i Q maxi, i N G, r minik r ik r maxik, r ik E disc discrete, ik T,

6 where: P ik = ν i 2 (y ik cos(ζ ik ) + g ik ) ν i ν k y ik cos(ζ ik + θ i θ k ), ik S e i, Q ik = ν i 2 (y ik sin(ζ ik ) h ik ) ν i ν k y ik sin(ζ ik + θ i θ k ), ik S e i, P ik = ν i 2 r ik 2 y ik cos(ζ ik ) ν i ν k r ik y ik cos(ζ ik + θ i θ k ), ik T e i, Q ik = ν i 2 r ik 2 y ik sin(ζ ik ) ν i ν k r ik y ik sin(ζ ik + θ i θ k ), ik T e i, P ki = ν k 2 (y ik cos(ζ ik ) + g ik ) ν i ν k y ik cos(ζ ik + θ k θ i ), ki S s i, Q ki = ν k 2 (y ik sin(ζ ik ) h ik ) ν i ν k y ik sin(ζ ik + θ k θ i ), ki S s i, P ki = ν 2 k (y ik cos(ζ ik ) + y 0 ik cos(ζ 0ik)) ν i ν k r ik y ik cos(ζ ik + θ k θ i ), ki Ti s, Q ki = ν 2 k (y ik sin(ζ ik ) + y 0 ik sin(ζ 0ik)) ν i ν k r ik y ik sin(ζ ik + θ k θ i ), ki Ti s. highly nonlinear, nonconvex

7 Problem Characteristics Nonconvex equality constraints with sin(w), cos(w), w 2, w 1 w 2 Bounds and network constraints Discrete variables: binary: a i (i N) variables on/off discrete: r ik E disc (ik T) r ik = NP ik N Sik (P) Nonconvex MINLP: minimize x,y g 0 (x, y), subject to g i (x, y) = 0, i = 1,.., m, (x, y) P, y Z t

8 Branch-and-bound for MINLP (binary variables) Drop the binary restrictions. Solve the continuous NLP. Continuous relaxation

9 Branch-and-bound for MINLP (binary variables) Choose a variable \ {0, 1}. Split the problem into 2 new subproblems. y i \{0, 1} y i = 0 y i = 1

10 Branch-and-bound for MINLP (binary variables) Choose a subproblem and solve it. Choose a variable \ {0, 1}. Split the problem into 2 new subproblems. y i \{0, 1} y y j \{0, 1}, i = 0 y i = 1 y j = 0 y j = 1

11 Branch-and-bound for MINLP (binary variables) The NLP is infeasible, cut the node. y i \{0, 1} y i = 0 y i = 1 y j = 0 y j = 1 infeasible problem

12 Branch-and-bound for MINLP (binary variables) If x NLP is feasible for MINLP, store U and cut the node. y i \{0, 1} y i = 0 y i = 1 y j = 0 y j = 1 discrete solution U

13 Branch-and-bound for MINLP (binary variables) Choose a subproblem and solve it. Choose a variable \ {0, 1}. Split the problem into 2 new subproblems. y i \{0, 1} y i = 0 y i = 1, y l \{0, 1} y l = 0 y l = 1

14 Branch-and-bound for MINLP (binary variables) The optimum value of NLP is larger than the current upper bound on MINLP. Cut the node. y i \{0, 1} y i = 0 y i = 1 y l = 0 y l = 1 U f

15 Branch-and-bound for MINLP (binary variables) Pursue like this until the whole tree has been explored. y i \{0, 1} y i = 0 y i = 1 y l = 1, y k \ {0, 1} y l =

16 Branch-and-bound for nonconvex problems: no guarantee to converge the continuous nonlinear relaxations are expensive to solve Our goal Find the global solution of the TVC problem by solving a sequence of appropriate linear problems

17 Overview Application & Motivation Tertiary Voltage Control Problem Characteristics Branch-and-Refine for Nonconvex MINLPs Special Ordered Set (SOS) Approximations Decomposition of Nonlinear Functions Piecewise Polyhedral Envelopes Branch-and-Refine Theoretical and Numerical Results Comparison to Underestimators & SOS-Branching Numerical Experience Conclusions & Future Research

18 Linear Approximation of a Nonconvex Function e.g.: sin(x)

19 Linear Approximation of a Nonconvex Function e.g.: sin(x) not accurate

20 Linear Approximation of a Nonconvex Function e.g.: sin(x) piecewise linear approximation

21 Special Ordered Set (SOS) Approximations (1 dimension) Choose n breakpoints x i and set f (x) f n (x) = λ i f (x i ) with n x = λ i x i i=1 i=1 n λ i = 1, λ i 0, i=1 i = 1 : n Refs: Beale, Tomlin

22 Special Ordered Set (SOS) Approximations (1 dimension) Choose n breakpoints x i and set f (x) f n (x) = λ i f (x i ) with n x = λ i x i i=1 i=1 n λ i = 1, λ i 0, i=1 i = 1 : n Refs: Beale, Tomlin

23 Special Ordered Set (SOS) Approximations (1 dimension) Choose n breakpoints x i and set f (x) f n (x) = λ i f (x i ) with n x = λ i x i i=1 i=1 n λ i = 1, λ i 0, i=1 i = 1 : n SOS condition: At most 2 λ i can be nonzero and these λ i must be adjacent Refs: Beale, Tomlin

24 SOS approximation method Construction of a linear approximation problem subject to SOS conditions satisfied via a branch-and-bound process Branching on the SOS variables λ i Used successfully by Martin et al. to solve nonconvex MINLP problems arising in gas network management Remark: in our case Relaxation of the SOS condition (No branching on SOS variables) SOS-like approx.

25 Infeasible SOS Approximations Solution of an approximation problem Physical constraints must be satisfied Equality constraints Solution has little chance to be feasible for our problem

26 Infeasible SOS Approximations Solution of an approximation problem Physical constraints must be satisfied Equality constraints Solution has little chance to be feasible for our problem

27 Infeasible SOS Approximations Solution of an approximation problem Physical constraints must be satisfied Equality constraints Solution has little chance to be feasible for our problem

28 Exponential Complexity of SOS Approximations approximate h(x, y) for (x, y) R n p breakpoints in each dimension p n SOS-variables λ i e.g., expression for real power has n = 8 variables impractical

29 Decomposition of Nonlinear Functions Idea: decompose h(x, y) into simpler functions: w j = x j j = 1,..., s, w s+j = y j j = 1,..., t, w s+t+j = h j (w j1 {, w j2 }) j = 1,..., K, where h j are univariate or bivariate and j 1, j 2 < s + t + j h(x, y) = w s+t+k Only 3 kinds of nonlinear h j functions for the TVC problem: square functions: x 2 trigonometric functions: sin(x), cos(x) bilinear functions: xy

30 Example Consider, for some constants a and b: h(x 1, x 2, x 3, x 4 ) = ax bx 2x 3 cos(x 4 ) x 1 w j = x j j = 1 : 4 w 5 = w 2 2 w 6 = w 2 w 3 w 7 = cos(w 4 ) w 8 = w 6 w 7 h = aw 5 + bw 8 w 1 Remark: decomposition not unique, e.g., w 6 = cos(w 4 ), etc.

31 Piecewise Polyhedral Envelopes Idea: Outer approximation by piecewise polyhedral envelopes

32 Piecewise Polyhedral Envelopes Idea: Outer approximation by piecewise polyhedral envelopes

33 Determination of an Envelope For each nonlinear function h (e.g., x 2, sin(x), cos(x), xy), determine the expression of the maximum approximation errors done by its SOS approximation: ɛ L,max = max x ( h(x) SOS h(x), 0), ɛ U,max = max x (h(x) h(x) SOS, 0), and approach h(x) by w h(x) where: h(x) ɛ L,max w h(x) h(x) + ɛ U,max

34 Approximation Errors Obtain maximum overestimation and underestimation errors of SOS approximation on [x k, x k+1 ]: Pre-computed bounds on [x k, x k+1 ], e.g., for x 2 : ɛ x2,l k = (x k+1 x k ) 2 4 and ɛ x2,u k = 0 (See Emilie s thesis for other functions)

35 Illustration with or without SOS Condition With SOS condition Without SOS condition

36 Piecewise Polyhedral Envelopes Surprise Theorem: Every (x, y, xy) with l x x u x and l y y u y is unique convex combination of (l x, l y, l x l y ), (l x, u y, l x u y ), (u x, l y, u x l y ) and (u x, u y, u x u y ), i.e. λ i 0, i = 1,..., 4: x y xy 1 = l x l x u x u x l y u y l y u y l x l y l x u y u x l y u x u y λ 1 λ 2 λ 3 λ 4 No need of envelope for xy!

37 Refinement of Approximations Piecewise envelope problem : (much) larger feasible domain than that of the original problem Need to refine the approximations Use of branch-and-bound to reduce the approximation intervals (the tighter the domain, the better the approximations) Use the same number of breakpoints Branch and Refine

38 Refinement of Approximations Piecewise envelope problem : (much) larger feasible domain than that of the original problem Need to refine the approximations Use of branch-and-bound to reduce the approximation intervals (the tighter the domain, the better the approximations) Use the same number of breakpoints Branch and Refine

39 Refinement of Approximations Piecewise envelope problem : (much) larger feasible domain than that of the original problem Need to refine the approximations Use of branch-and-bound to reduce the approximation intervals (the tighter the domain, the better the approximations) Use the same number of breakpoints Branch and Refine Ideal framework for discrete problems Convergence to the global solution (within some accuracy) is guaranteed under mild assumptions

40 Branch-and-Refine Solve piecewise envelope problem Branch on continous variables or integer variables Refine and tighten the envelope while going down the tree Exploit exactness of bilinear terms

41 Branch-and-Refine: Branching 1 dimension 2 dimension Illustration of branching and refinement

42 Branch-and-Refine: Fathoming Rules Assume at node k: x X k and y Y k (continuous domains) LP(X k, Y k ): LP relaxation of piecewise envelope problem NLP(X k, Y k ): z NLPk := minimize x,y subject to g 0 (x, y) g i (x, y) = 0, i = 1,.., m x X k, y Y k Fathoming Rules: 1. Infeasible LP(X k, Y k ) relaxation 2. NLP(X k, Y k ) solution same as LP(X k, Y k ) relaxation 3. LP(X k, Y k ) relaxation dominated by incumbent

43 Branch-and-Refine: Algorithm set U =, k = 1 & put LP(X k, Y k ) on stack while stack is not empty solve LP(X k, Y k ) solution (x k, y k ) if LP(X k, Y k ) infeasible or z LPk U ɛ then fathom node (Case 1. or 3.) else solve NLP(X k, Y k ) solution (ˆx k, ŷ k ) if z NLPk < U ɛ & ŷ k integer then update U := z NLPk & incumbent (x, y ) := (ˆx k, ŷ k ) if z NLPk z LPk ɛ then fathom node (Case 2.) else branch (creating two new LPs) and refine k = k+1

44 Overview Application & Motivation Tertiary Voltage Control Problem Characteristics Branch-and-Refine for Nonconvex MINLPs Special Ordered Set (SOS) Approximations Decomposition of Nonlinear Functions Piecewise Polyhedral Envelopes Branch-and-Refine Theoretical and Numerical Results Comparison to Underestimators & SOS-Branching Numerical Experience Conclusions & Future Research

45 Comparison to Caratzoulas & Floudas [Caratzoulas & Floudas, 2004] construct convex underestimators: ( ) 1 U sin (x) = sin (x + 2π) , 6 and ( ) 1 U cos (x) = sin (x + 2π) Theorem: Let A sin SOS, Acos SOS be SOS underestimation errors and Asin trig, Acos trig be underestimation errors of [C&F, 2004]. Then A sin SOS = Asin trig and A cos SOS = Acos trig piecewise polyhedra are tighter on [0, 2π].

46 Comparison to Caratzoulas & Floudas: sin(x)

47 Comparison to Caratzoulas & Floudas: cos(x)

48 Comparison to SOS-Branching Example: x 2 Better to refine than to branch on SOS variables (factor 11/12)

49 Test Problems (Generic) prob #var #cons #var OA #cons OA #sets λ #disc pb pb pb pb pb pb pb pb pb pb pb pb pb pb

50 Test Problems (Tertiary Voltage Control) prob #var #cons #var OA #cons OA #sets λ #disc TVC TVC TVC TVC TVC TVC moderately sized problems

51 Comparison with Other Softwares Available on NEOS Nonlinear global optimization solvers available on NEOS: BARON: not directly applicable due to sin(x), cos(x) Possibility: replace each trigonometric function by its Taylor approximation of degree 7 (Bussieck) but not accurate Nonlinear local optimization solvers available on NEOS: + cheaper than our method since local methods - no guarantee to converge to the global optimum

52 Comparison with Local Methods For NLP: Branch-and-Refine IPOPT KNITRO FilterSQP # solved # global solution For MINLP: Branch-and-Refine Bonmin MINLP BB # solved # global solution

53 Implementation Details & Tricks LPs solved with CPLEX decomposition (hand-coded by Emilie) exploit common sub-expressions can be automated, similar to automatic differentiation (AD) NLPs solved with FilterSQP (AD for gradients/hessians) propagate & strengthen bounds through computational graph pre-solve (LP) to reduce range of variables (like BARON) adaptive presolve is best: trail-off factor (generalized) pseudo-cost branching (generalized) best-estimate node selection

54 Numerical Results (# LPs solved) prob basic +presolve +var-select +node-select pb pb pb pb pb pb pb pb pb8 fail pb pb pb pb12 fail pb13 fail

55 Numerical Results (# LPs solved) prob basic +presolve +var-select +node-select TVC TVC2 fail TVC TVC4 fail TVC5 fail TVC6 fail

56 Conclusions & Future Research Branch-and-Refine Three key ingredients: 1. decompose functions into 1D and 2D components 2. construct piecewise polyhedral envelope 3. branch on (continuous or integer) variables, not on SOS ones favorable theoretical properties encouraging numerical results Future Work exploit expression tree... use AD tricks for better OA what decomposition is best... non-unique avoid λ variables... work with OA directly avoid SOS approx. for convex functions... NLP subproblems efficient implementation & support for AMPL, GAMS