The R-to-MOSEK Optimization Interface Henrik Alsing Friberg MOSEK ApS, Fruebjergvej 3, Box 16, DK 2100 Copenhagen, Blog: http://themosekblog.blogspot.com ISMP, Berlin 2012 http://www.mosek.com
Introduction 2 / 48
What is R? Introduction The R Project What is R? What is MOSEK? A fast open-source programming language for What is Rmosek? Installation technical computing and graphics. Simple models Advanced models Highlights: Better formulations Data handling and visualization ■ One million users – Intel Capital, 2009 High performance Summary ■ The Comprehensive R Archive Network (3895 packages and counting) ■ Direct interfaces to C and Fortran (as well as C++, Perl, Java, Python, Matlab, Excel etc.) ■ Revolution Analytics ■ RStudio 3 / 48
What is MOSEK? Introduction MOSEK What is R? What is MOSEK? A high performance software library designed to What is Rmosek? Installation solve large-scale convex optimization problems. Simple models Advanced models Highlights: Better formulations Data handling and visualization ■ Linear constraints (LP) High performance Summary ■ Second-order cone constraints (SOCP) ■ Semidefinite constraints (SDP) ■ Mixed-integer variables (MIP) ■ Quadratic terms (QP and QCQP) ■ Separable convex operators – log( x i ) , exp( x i ) , ... 4 / 48
What is Rmosek? Introduction Rmosek What is R? What is MOSEK? A simple interface from R to the optimizers of What is Rmosek? Installation MOSEK. Simple models Advanced models Why? Better formulations Data handling and visualization ■ Some financial customers find R better than MATLAB. High performance Summary ■ Replace unofficial implementations. 5 / 48
What is Rmosek? Introduction Rmosek What is R? What is MOSEK? A simple interface from R to the optimizers of What is Rmosek? Installation MOSEK. Simple models Advanced models Why? Better formulations Data handling and visualization ■ Some financial customers find R better than MATLAB. High performance Summary ■ Replace unofficial implementations. Goals: ■ Open-source ■ Highly effective ■ Integrate with what R users normally do 5 / 48
Installation Introduction A oneliner for Windows, MacOS and Linux: What is R? install.packages("Rmosek", type="source") What is MOSEK? What is Rmosek? Installation Simple models Advanced models Better formulations Data handling and visualization High performance Summary 6 / 48
Installation Introduction A oneliner for Windows, MacOS and Linux: What is R? install.packages("Rmosek", type="source") What is MOSEK? What is Rmosek? Installation Package overview: Simple models library(help="Rmosek") Advanced models Better formulations Data handling and Major functions: visualization High performance ■ mosek(prob) Summary ■ mosek write(prob, file) ■ mosek read(file) The problem is an R-variable.. Create as you like! 6 / 48
Simple models 7 / 48
Linear optimization Introduction Simple models maximize 3 x 1 + 1 x 2 + 5 x 3 + 1 x 4 Linear optimization 3 x 1 + 1 x 2 + 2 x 3 = 30 , subject to Advanced models 2 x 1 + 1 x 2 + 3 x 3 + 1 x 4 ≥ 15 , Better formulations 2 x 2 + 3 x 4 ≤ 25 , Data handling and visualization High performance 0 ≤ ≤ ∞ , x 1 Summary 0 ≤ ≤ 10 , x 2 0 ≤ x 3 ≤ ∞ , 0 ≤ x 4 ≤ ∞ . 8 / 48
Linear optimization Introduction lo1 <- list() Simple models Linear optimization ## Objective Advanced models lo1$sense <- "max" Better formulations Data handling and lo1$c <- c(3, 1, 5, 1) visualization High performance ## Constraint matrix Summary lo1$A <- spMatrix(nrow = 3, ncol = 4, i = c(1, 2, 1, 2, 3, 1, 2, 2, 3), j = c(1, 1, 2, 2, 2, 3, 3, 4, 4), x = c(3, 2, 1, 1, 2, 2, 3, 1, 3) ) 9 / 48
Linear optimization Introduction ## Constraint bounds Simple models lo1$bc <- cbind(c(30, 30), Linear optimization c(15, Inf), Advanced models c(-Inf, 25)) Better formulations Data handling and visualization ## Variable bounds High performance lo1$bx <- cbind(c(0, Inf), Summary c(0, 10), c(0, Inf), c(0, Inf)) 10 / 48
Advanced models 11 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = 1 . 0 , optimization Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , 12 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = 1 . 0 , optimization Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , sdo1 <- list() sdo1$sense <- "minimize" sdo1$c <- c(1,0,0) 13 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = 1 . 0 , optimization Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , sdo1$A <- spMatrix(nrow=2, ncol=3, i = c(1,2,2), j = c(1,2,3), x = c(1,1,1) ) 14 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = optimization 1 . 0 , Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , sdo1$bc <- cbind(c(1.0, 1.0), c(0.5, 0.5)) 15 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = 1 . 0 , optimization Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , sdo1$bx <- cbind(c(-Inf, Inf), c(-Inf, Inf), c(-Inf, Inf)) 16 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = 1 . 0 , optimization Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , sdo1$cones <- cbind( list("quad", c(1,2,3)) ) 17 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = 1 . 0 , optimization Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , sdo1$bardim <- c(3) 18 / 48
Second-order cone and semidefinite optimization 2 1 0 Introduction • x 1 + x 1 1 2 1 minimize Simple models 0 1 2 Advanced models Second-order cone 1 0 0 and semidefinite • x 1 + x 1 subject to 0 1 0 = 1 . 0 , optimization Better formulations 0 0 1 Data handling and 1 1 1 visualization • x 1 + x 2 + x 3 1 1 1 = 0 . 5 , High performance 1 1 1 Summary � ( x 1 , x 2 , x 3 ) ∈ R 3 , x 2 2 + x 2 x 1 ≥ x 1 � 0 . 3 , # Block triplet format (SDPA) sdo1$barc$j <- c(1,1,1,1,1) sdo1$barc$k <- c(1,2,2,3,3) sdo1$barc$l <- c(1,1,2,2,3) sdo1$barc$v <- c(2,1,2,1,2) 19 / 48
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