Package splitfeas. April 11, 2018

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1 Title Multi-Set Split Feasibility Version Package splitfeas April 11, 2018 An implementation of the majorization-minimization (MM) algorithm introduced by Xu, Chi, Yang, and Lange (2017) <arxi: > for soling multiset split feasibility problems. In the multi-set split feasibility problem, e seek to find a point in the intersection of multiple closed sets and hose image under a mapping also must fall in the intersection of seeral closed sets. Depends compiler, corpcor, matristats License MIT + file LICENSE RoygenNote NeedsCompilation no Author Eric C. Chi [aut, cre, cph], Jason Xu [ctb], Meng Yang [ctb], Kenneth Lange [ctb] Maintainer Eric C. Chi <ecchi1105@gmail.com> Repository CRAN Date/Publication :15:34 UTC R topics documented: backtrack ddg dg mmqn_step nmsfp_mm nmsfp_mmqn nmsfp_sap nmsfp_sap_one_step project_ball project_cube project_halfspace

2 2 ddg project_square proimity qnamm softma split_feasibility ood_in_sole Inde 13 backtrack Backtracking Line Search Gien a descent direction backtrack computes a step size that ensures sufficient decrease in an objectie. backtrack(, d, f,, alpha = 0.01, beta = 0.8) d f alpha beta Current iterate Descent direction objectie function gradient of objectie function sufficient decrease parameter sufficient decrease parameter ddg Compute the approimate Hessian of the majorization. ddg computes the Hessian of the majorization of the proimity function. ddg(,,, )

3 dg 3 non-anchor point Eamples set.seed(12345) n <- 10 p <- 2 <- matri(rnorm(p),p,1) <- 1 <- 1 A <- matri(rnorm(n*p),n,p) <- function() {return(t(a))} sol <- ddg(,,,) dg Compute the gradient of the majorization. dg computes the gradient of the majorization of the proimity function. dg(, a,,, plist1, plist2, h, ) a plist1 plist2 h non-anchor point Anchor point list of projection functions for first set of constraints; each takes a single point list of projection functions for second set of constraints; each takes a single point Function handle for output mapping

4 4 nmsfp_mm mmqn_step MM-quasi-Neton step mmqn_step computes a single step. mmqn_step(,,, plist1, plist2, f,, h,, oodbury = TRUE) plist1 plist2 f h oodbury Current iterate list of projection functions for first set of constraints; each takes a single point list of projection functions for second set of constraints; each takes a single point objectie function gradient of objectie function Function handle for output mapping Boolean: TRUE to use the Woodbury inersion formula nmsfp_mm MM algorithm for nonlinear multiple-sets split feasibility problem nmsfp_mm uses quasi-neton updates to sole the nonlinear multiple-sets split feasibility problem. nmsfp_mm(0,,, plist1, plist2, f,, h,, tol = 1e-10, ma_iter = 1000)

5 nmsfp_mmqn 5 0 plist1 plist2 f h tol ma_iter Initial iterate list of projection functions for first set of constraints; each takes a single point list of projection functions for second set of constraints; each takes a single point objectie function gradient of objectie function Function handle for output mapping Stopping tolerance Maimum number of iterations See Also mmqn_step nmsfp_mmqn MM algorithm (accelerated) for nonlinear multiple-sets split feasibility problem nmsfp_mmqn uses quasi-neton updates to sole the nonlinear multiple-sets split feasibility problem. nmsfp_mmqn(0,,, plist1, plist2, f,, h,, qn = 5, tol = 1e-10, ma_iter = 1000) 0 plist1 plist2 Initial iterate list of projection functions for first set of constraints; each takes a single point list of projection functions for second set of constraints; each takes a single point

6 6 nmsfp_sap f h qn tol ma_iter objectie function gradient of objectie function Function handle for output mapping number of secants conergence tolerance maimum number of iterations See Also mmqn_step, qnamm nmsfp_sap Self-adaptie projection-type method algorithm for nonlinear multiple-sets split feasibility problem nmsfp_sap performs the self-adaptie projection-type method of Li et al. nmsfp_sap(0,,, plist1, plist2, proj, f,, h,, delta = 0.3, mu = 0.7, beta0 = 1, tol = 1e-10, ma_iter = 1000) 0 plist1 plist2 proj f h delta mu beta0 tol ma_iter Initial iterate list of projection functions for first set of constraints; each takes a single point list of projection functions for second set of constraints; each takes a single point handle to projection operation. objectie function gradient of objectie function Function handle for output mapping step-size parameter step-size parameter initial Tolerance Maimum number of iterations

7 nmsfp_sap_one_step 7 nmsfp_sap_one_step One step of self-adaptie projection-type method for the NMSFP nmsfp_sap_one_step performs the self-adaptie projection-type method of Li et al. nmsfp_sap_one_step(, delta, mu, tau, gamma,, proj) delta mu tau gamma proj current iterate step-size parameter step-size parameter step-size parameter step-size parameter handle to gradient of objectie function handle to projection operation. References Li, Han, and Zhang. (2013) A self-adaptie projection-type method for nonlinear multiple-sets split feasibility problem, Inerse Problems in Science and Engineering, 21(1): project_ball Projection onto a ball project_ball computes the Euclidean projection of a point onto a ball. project_ball(, center, r) center r Point to project Center of the sphere Radius of the sphere

8 8 project_halfspace Eamples set.seed(12345) p <- 3 center <- rnorm(p) r <- runif(1) <- rnorm(p) y <- project_ball(,center,r) project_cube Project onto a cube project_cube computes the Euclidean projection of a point onto a cube. project_cube(, center, r) center r Point to project Center of the square Half the length of a side Eamples set.seed(12345) p <- 3 center <- matri(rnorm(p),p,1) r <- runif(1) <- matri(rnorm(p),p,1) y <- project_cube(,center,r) project_halfspace Projection onto a halfspace project_halfspace computes the Euclidean projection of a point onto a closed half-space. The function returns the projection onto the set project_halfspace(, a, b)

9 project_square 9 a b Point to project is the normal ector is the threshold Eamples set.seed(12345) p <- 3 a <- matri(rnorm(p),p,1) a <- a/norm(a,'f') b <- runif(1) <- matri(rnorm(p),p,1) y <- project_halfspace(,a,b) project_square Project onto a square project_square computes the Euclidean projection of a point onto a square. project_square(, center, r) center r Point to project Center of the square Half the length of a side Eamples set.seed(12345) p <- 2 center <- matri(rnorm(p),p,1) r <- runif(1) <- matri(rnorm(p),p,1) y <- project_square(,center,r)

10 10 qnamm proimity Proimity function proimity computes the proimity function. proimity(,,, plist1, plist2, h) plist1 plist2 h Current iterate list of projection functions for first set of constraints; each takes a single point list of projection functions for second set of constraints; each takes a single point Function handle for output mapping qnamm Quasi-Neton acceleration of MM algorithm qnamm performs Quasi-Neton acceleration of an MM algorithm. qnamm(, f_mm, qn, f_obj, ma_iter = 100, tol = 1e-06,...) initial iterate f_mm MM algorithm map qn number of secants f_obj handle to objectie function ma_iter maimum number of iterations tol conergence tolerance... Additional arguments to pass to f_mm

11 softma 11 References H Zhou, D Aleander, and K Lange. (2011) A quasi-neton acceleration method for high-dimensional optimization algorithms, Statistics and Computing, 21(2): softma Compute soft-ma softma returns the soft maimum of a collection of reals. softma(, a = 100) a input scaling factor Eamples set.seed(12345) n <- 10 <- rnorm(n) softma() split_feasibility split_feasibility split_feasibility

12 12 ood_in_sole ood_in_sole Compute the inerse approimate Hessian of the majorization using the Woodbury inersion formula. ood_in_sole computes the inerse of the Hessian term of the majorization of the proimity function using the Woodbury formula. The function mmqn_step inokes ood_in_sole instead of ddg if the argument oodbury=true. This should be used hen p «n. Compute the inerse approimate Hessian of the majorization using the Woodbury inersion formula. ood_in_sole computes the inerse of the Hessian term of the majorization of the proimity function using the Woodbury formula. The function mmqn_step inokes ood_in_sole instead of ddg if the argument oodbury=true. This should be used hen p «n. ood_in_sole(,,,, ) non-anchor point Right hand side

13 Inde backtrack, 2 ddg, 2 dg, 3 mmqn_step, 4 nmsfp_mm, 4 nmsfp_mmqn, 5 nmsfp_sap, 6 nmsfp_sap_one_step, 7 project_ball, 7 project_cube, 8 project_halfspace, 8 project_square, 9 proimity, 10 qnamm, 10 softma, 11 split_feasibility, 11 split_feasibility-package (split_feasibility), 11 ood_in_sole, 12 13