VIEWS: 7 PAGES: 12 CATEGORY: Education POSTED ON: 2/13/2010
Interior-point algorithms for linear programs with many inequality constraints Luke Winternitz (lukewinternitz@gmail.com) ´ Joint work with (advisor) Andre Tits, P.A. Absil, D.P. O’Leary April 26, 2008 Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 1 / 12 Primal-dual interior-point algorithms for LP Consider the primal and dual standard LP forms min c T x max b T y s.t. Ax = b s.t. AT y ≤ c x ≥0 (or s.t. AT y + s = c, s ≥ 0) where A ∈ Rm×n and we assume n ≫ m. Primal-dual interior point (PDIP) methods apply Newton’s method to the perturbed KKT optimality conditions AT c − AT y − s 0 10 1 0 1 0 I ∆x AT y + s = c, @A 0 0 A @∆y A = @ b − Ax A , Ax = b, S 0 X ∆s σµe − Xs xi si = τ, i ∈ n T where τ = σµ, µ = x n s > 0, σ ∈ [0, 1], (x, s) ≥ 0. X := diag(x) > 0, S := diag(s) > 0, e := (1, 1, · · · 1)T . Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 2 / 12 Cost of an iteration Commonly, the Newton-KKT system is reduced (by block gaussian elimination) to to the positive deﬁnite “normal-equations” AS −1 XAT ∆y = (⋆). Dominant cost is O(nm2 ), for forming the “normal-matrix” n X xi AS −1 XAT = ai aT . i si i=1 Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 3 / 12 The case of n ≫ m: many inequality constraints in the dual We expect many constraints are redundant or somehow not very relevant. We could try to guess a good set to “pay attention to” and ignore the rest. irrelevant? b b ∆y ∆y redundant active Ignore many constraints max b T y max b T y s.t. AT y ≤ cQ Q s.t. AT y ≤ c (AQ := [ai1 , ai2 , · · · ], ∀ ij ∈ Q) Some prior work in 1990’s, Dantzig and Ye [1991], Tone [1993], Den Hertog [1994], but we think there is further room to explore. Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 4 / 12 Constraint-reduced PDIP method Given a small set Q of critical constraints, compute the PDIP direction for the reduced primal-dual pair T min cQ xQ max b T y s.t. AQ xQ = b s.t. AT y ≤ cQ Q xQ ≥ 0 T xQ sQ i.e. let µQ = |Q| and solve 0 AT 0 10 1 0 1 Q IQ ∆xQ 0 @ AQ 0 0 A @ ∆y A = @ b −AQ xQ A , SQ 0 XQ ∆sQ σµQ −XQ sQ This deﬁnes a partial search direction (∆xQ , ∆y, ∆sQ ) which is cheaper to compute than the full PDIP direction. −1 X xi AQ SQ XQ AT := Q ai aT i costs O(|Q|m2 ) vs. O(nm2 ) si i∈Q Issues: How to update (x, y, s) using this partial direction? (next slide). How to select Q? Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 5 / 12 A possible approach: reduced normal matrix approximation (Tits et al. [2006]) If we solve normal equations, e.g. in afﬁne-scaling (σ = 0), dual-feasible case: AS −1 XAT ∆y = b ∆s = −AT ∆y (#) ∆x = −x − S −1 X ∆s. One possibility is to simply make the substitution: −1 AQ SQ XQ AT −→ AS −1 XAT , Q in (#) above, and change nothing else. Then the search direction is fully deﬁned. Other methods for ﬁlling out the update of (x, y, s) are possible. Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 6 / 12 Choosing the constraint set Q, and a convergence result Minimal requirements on Q: AQ is full row rank, Q contains the m most nearly active constraints, Theorem: Given this and under a nondegeneracy assumption, we have global convergence with a local quadratic rate of two variants of the reduced PDIP method: Afﬁne-scaling (Tits et al [2006]) (Mehrotra) predictor-corrector (Winternitz et al [2007]) However, additional constraints may be needed for good numerical performance. Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 7 / 12 Removing a strong assumption. Requirement 1: “AQ is full row rank” is One approach: detect the degenerate troublesome. situation and take a ”kernel step” b S tep a1 a1 el rn Ke a2 a2 Alternatively, regularize the reduced problem 2 max b T y − δk y a1 Step is now well deﬁned s.t. AT y ≤ cQ Q a2 Theorem: Under appropriate assumptions, the convergence result again holds for a kernel-step variant, and a regularized variant that uses δk = O(µk ). Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 8 / 12 Random problems m = 200, n = 40000 Generate: A, b, y0 ∼ N (0, 1), s0 ∼ U(0, 1). Normalize columns of A. Set c = AT y0 + s0 and x0 = e. CPU time Iterations We can evidently choose |Q| ≈ m here. This implies an asymptotic (large n) savings of a factor of m. Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 9 / 12 Discretized semi-inﬁnite linear programs Semi-inﬁnite linear programming (SILP) is a very powerful tool for engineering design. Applications include design of, e.g., digital ﬁlters, antenna arrays, and control systems. Can (approximately) solve SILPs by discretizing the constraint set to obtain a ﬁnite linear program (with n ≫ m). Example: Minimax FIR ﬁlter design. min t ˛ ˛ s.t. ˛H(e jω , b) − Hd (e jω )˛ ≤ t ˛ ˛ ω ∈ [0, π] Posed as LP of size 201 × 20000. Cost to solve: Standard PDIP (MPC): 34 iterations, 61 seconds. Reduced PDIP (MPC): 45 iterations, 8 seconds. Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 10 / 12 A possible control application: LP formulation of Markov decision processes (MDP) A simple version of the (discounted, inﬁnite-horizon) MDP is to solve "∞ # X k min J(x0 , µ) = E α g(xk (µ)) µ k =1 {xk } a DTFS controlled Markov chain, with states in X = {1, 2, . . . p} and controls in U = {1, 2, . . . q} and µ is a policy (a map from states to controls). LP formulation: Using cost approx. J = Hr (r ∈ Rk ) in (∗), get “approximate LP” (ALP) formulation: max e T J max (H T c)T r s.t. J ≤ g + αPu J (∗) s.t. (Hr ) ≤ g + αPu Hr for all u ∈ U. for all u ∈ U. p variables and pq constraints k variables and pq ≫ k constaints. Reduced PDIP methods may be useful for this class of problems. Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 11 / 12 Last slide Summary: We proposed a variation of the PDIP method for solving LPs with many inequality constraints. Basic idea is to use small subset of constraints to construct search direction. Proved global and local quadratic convergence of several variants with only requirement that the m most nearly active constraints are always in Q. Numerical results are promising. Future work: Take a closer look at applications. Extension to convex programming, general nonlinear-programming. Conceptually, there is no problem if the constraints are bent outward or inward. ∇f ( y) Winternitz et al. (UMCP) IPMs w/ many constraints April 26, 2008 12 / 12