Mixed Integer Second Order Cone Optimization (MISOCO): Cuts, Warm Start, and Rounding

Transcription

1 Mixed Integer Second Order Cone Optimization (MISOCO): Conic Tamás Terlaky, Mohammad Shahabsafa, Julio C. Góez, Sertalp Cay, Imre Pólik OPDGTP, Tel Aviv, Israel April /35

2 Outline 1 Disjunctive Conic Cuts (DCCs) for MISOCO MISOCO DCCs MISOCO Solution Approaches /35

3 Outline Disjunctive Conic Cuts (DCCs) for MISOCO MISOCO DCCs MISOCO Solution Approaches 1 Disjunctive Conic Cuts (DCCs) for MISOCO MISOCO DCCs MISOCO Solution Approaches /35

4 Conic Optimization MISOCO DCCs MISOCO Solution Approaches A general conic optimization (CO) problem is defined as min c, x s.t. a i, x = b i x K K denotes a closed pointed convex cone c, x denotes the inner product of vectors c and x. - K = R n + : Linear Optimization (LO) - K = L n 1 L n 2 L nm : Second Order Cone Optimization (SOCO) - K = S n + : Semidefinite Optimization (SDO) In this case c, x, and a i are symmetric matrices, and c, x = Tr(cx). 4/35

5 The MISOCO problem MISOCO DCCs MISOCO Solution Approaches Mixed Integer Second-Order Conic Optimization (MISOCO) problems min c T x s.t. Ax = b x L x Z d R n+1 d, in which L is the Cartesian product of second-order cones. For simplicity, we assume that L is a single second-order cone. A Second-Order Cone (SOC) is defined as follows L n+1 = {x = (x 0, x 1,..., x n) (x 1, x 2,..., x n) 2 x 0 }. MISOCO problems can be solved using a branch and cut methodology. We can add cuts to strengthen the formulation and reduce the solution time. Nonlinear cuts for MISOCO problems have recently received attention 5/35

6 Disjunction on a convex set MISOCO DCCs MISOCO Solution Approaches Let X R n, n > 1 be a full dimensional closed convex set. Consider two half-spaces A = {x R n : a T x α} B = {x R n : b T x β}, where a, b R n and (a T, α), (b T, β) are not scalar multiple of each other. Assumptions: The intersection A B X is empty. The intersections X A = and X B = are nonempty. Disjunctive Conic Cut (DCC): A closed convex cone K R n with dim(k) > 1 is called a DCC for X and the disjunction A B if conv(x (A B) = X K. DCCs always exist for MISOCO problems (Belotti et al.). 6/35

7 MISOCO DCCs MISOCO Solution Approaches Illustration of a disjunctive conic cut for a MISOCO problem Illustration of DCC for a MISOCO problem The figure is from Góez 7/35

8 Uni-parametric family of quadrics MISOCO DCCs MISOCO Solution Approaches Definition 1 Let P R l l, p, w R l and ρ R, then the quadric Q is the set defined as Q = {w R l w P w + 2p w + ρ 0}. Theorem 2 Let (P, p, ρ) be a quadric and consider two hyperplanes A = = {z a z = α} and B = = {z d z = β}. The family of quadrics (P (τ), p(τ), ρ(τ)) parameterized by τ R b having the same intersection with A = and B = as the quadric (P, p, ρ) is given by P (τ) = P + τ adt + da T 2 βa + αd p(τ) = p τ 2 ρ(τ) = ρ + ταβ. 8/35

9 MISOCO Solution Approaches MISOCO DCCs MISOCO Solution Approaches Linear Approximation of SOCOs ; (GaTech: Nemhauser, Savelsberg,...; Lehigh: Ralphs, Bulut) ++ Allows to use advanced MILO methodology Inferior for MISOCO with higher dimensional cones Use IPMS fof MISOCO with B&DCCs (Lehigh and descendants) ++ New powerful cuts, power of IPMs for SOCO Novel methodology Conic-MILO methodology needed 9/35

10 Outline 1 Disjunctive Conic Cuts (DCCs) for MISOCO MISOCO DCCs MISOCO Solution Approaches /35

11 Mean variance portfolio optimization model Round Lot constraints minimize: x Σx subject to: µ 0 x 0 + µ x r x 0 + e x = 1 x i = a i z i i = 1,..., n 0 x i 1 i = 1,..., n z Z n +, (1) Denote ˆΣ = diag(a) Σdiag(a). Denote ˆµ = diag(µ)a + µ 0 a, and ˆr = µ 0 r 11/35

12 Send quadratic objective function to the constraint and define new variable t Revised RL-MVPO minimize: t subject to: ˆµ z ˆr a z 1 0 a i z i 1 z ˆΣz t z Z N + (RL-MVPO) 12/35

13 Comparison of solution approaches For round lot problems, we compared BB, BCC-I and MOSEK Number of nodes Solution Time Data BB BCC-I MOSEK BB BCC-I MOSEK AA RD RD RD RD RD RD RD RD RD RD Comparison of number of nodes and solution time of solution approaches for round-lot AAPs. 13/35

14 Outline 1 Disjunctive Conic Cuts (DCCs) for MISOCO MISOCO DCCs MISOCO Solution Approaches /35

15 Pathological Disjunctions See Julio Góez s presentation! 15/35

16 Outline Disjunctive Conic Cuts (DCCs) for MISOCO 1 Disjunctive Conic Cuts (DCCs) for MISOCO MISOCO DCCs MISOCO Solution Approaches /35

17 Rounding - Jordan Frames 17/35

18 Jordan Frame LO Let x i L ni, then the the eigenvalues and Jordan vectors are given as: λ + i = x i 1 + xi 2:n i λ i = x i 1 xi 2:n i f + i = x i 2:n i x i 2:n i f i = xi 2:n i x i 2:n i and we have: x i = λ + i f + i + λ i f i. 18/35

19 Jordan Frame LO 19/35

20 The Rounding LO Problem 20/35

21 Illustration: Primal and Dual Rounding 21/35

22 Duality in Rounding Schemes 22/35

23 Primal Penalty and Rounding problems 23/35

24 Primal Rounding Algorithm 24/35

25 Flow of Primal Rounding Algorithm 25/35

26 Dual Penalty and Rounding problems 26/35

27 Dual Rounding Algorithm 27/35

28 Flow of Dual Rounding Algorithm 28/35

29 Test Problem Set 29/35

30 Results Primal and Dual Rounding 30/35

31 Results Primal-Dual adn Hybrid Rounding 31/35

32 Methodology 32/35

33 Numerical Experiences Proof of the pie 33/35

34 Conclusion Disjunctive Conic Cuts (DCCs) for MISOCO We presented Disjunctive Conic Cuts (DCCs) for MISOCO. We demonstrated the power of DCCs. The identification of the pathological cases is important for the efficient implementation of DCCs (see Góez). Utilized Jordan frames to developed Primal, Dual, and Prima-Dual Hybrid rounding heuristics for MISOCOs. Developed an efficient warm-start method for SOCO. Both the rounding and warm-start methodologies are proved to be efficient. 34/35

35 Thanks Any questions? 35/35