SCS = splitting conic solver
SCS is a numerical optimization package for solving large-scale convex cone problems, written in C, based on our paper Operator Splitting for Conic Optimization via Homogeneous Self-Dual Embedding.
SCS can be used in other C, C++, Python, Matlab, Julia, Java, and Scala programs via included interfaces. (Julia interface available here.)
SCS can be called from parser-solvers CVX, CVXPY, Convex.jl, and Yalmip.
SCS numerically solves convex cone programs using the alternating direction method of multipliers (ADMM). It returns solutions to both the primal and dual problems if the problem is feasible, or a certificate of infeasibility otherwise. SCS solves the following primal cone problem:
minimize c'x
subject to Ax + s = b
s in K
over variables x
and s
, where A
, b
and c
are user-supplied data and K
is a user-defined convex cone.
The dual problem is given by
maximize -b'y
subject to -A'y == c
y in K^*
over variable y
, where K^*
denotes the dual cone to K
.
The cone K
can be any Cartesian product of the following primitive cones:
+ zero cone {x | x = 0 }
(dual to the free cone {x | x in R}
)
+ positive orthant {x | x >= 0}
+ second-order cone {(t,x) | ||x||_2 <= t}
+ positive semidefinite cone { X | min(eig(X)) >= 0, X = X^T }
+ exponential cone {(x,y,z) | y e^(x/y) <= z, y>0 }
+ dual exponential cone {(u,v,w) | −u e^(v/u) <= e w, u<0}
+ power cone {(x,y,z) | x^a * y^(1-a) >= |z|, x>=0, y>=0}
+ dual power cone {(u,v,w) | (u/a)^a * (v/(1-a))^(1-a) >= |w|, u>=0, v>=0}
The rows of the data matrix A
correspond to the cones in K
.
The rows of A
must be in the order of the cones given above, i.e., first come the
rows that correspond to the zero/free cones, then those that correspond to the
positive orthants, then SOCs, etc. For a k
dimensional semidefinite cone
when interpreting the rows of the data matrix A
SCS assumes that the k x k
matrix variable has been vectorized by stacking the
lower triangular elements column-wise to create a vector of length k(k+1)/2
.
At termination SCS returns solution (x*, s*, y*)
if the problem is feasible,
or a certificate of infeasibility otherwise. See here
for (much) more details about cone programming and certificates of infeasibility.
Linear equation solvers
Each iteration of SCS requires the solution of a set of linear equations. This package includes two implementations for solving linear equations: a direct solver which uses a cached LDL factorization and an indirect solver based on conjugate gradients.
The direct solver uses the LDL and AMD packages numerical linear algebra packages by Timothy Davis and others, the necessary files are included. See here for more information about these packages.
Typing make
at the command line will compile the code and create
SCS libraries in the out
folder. It will also produce four demo binaries in the
out
folder named demo_direct
, demo_indirect
, demo_SOCP_direct
and
demo_SOCP_indirect
.
If make
completes successfully, it will produce two static library files,
libscsdir.a
and libscsindir.a
under the out
folder and two dynamic
library files libscsdir.ext
and libscsindir.ext
(where .ext
extension is
platform dependent) in the same folder. To include the
libraries in your own source code, compile with the linker option with
-L(PATH_TO_SCS)\lib
and -lscsdir
or -lscsindir
(as needed).
These libraries (and scs.h
) expose only four API functions:
Work * scs_init(const Data * d, const Cone * k, Info * info);
This initializes the Work struct containing the workspace that scs will
use, and performs the necessary preprocessing (e.g. matrix factorization).
All inputs d
, k
, and info
must be memory allocated before calling.
scs_int scs_solve(Work * w, const Data * d, const Cone * k, Sol * sol, Info * info);
This solves the problem as defined by Data d
and Cone k
using the workspace
in w
. The solution is returned in sol
and information about the solve
is returned in info
(outputs must have memory allocated before calling).
None of the inputs can be NULL. You can call scs_solve
many times for one
call to scs_init
, so long as the matrix A
does not change (b and c can change).
void scs_finish(Work * w);
Called after all solves completed to free allocated memory and other cleanup.
scs_int scs(const Data * d, const Cone * k, Sol * sol, Info * info);
Convenience method that simply calls all the above routines in order, for cases where the workspace does not need to be reused. All inputs must have memory allocated before this call.
The relevant data structures are:
typedef struct SCS_PROBLEM_DATA Data;
typedef struct SCS_SETTINGS Settings;
typedef struct SCS_SOL_VARS Sol;
typedef struct SCS_INFO Info;
typedef struct SCS_SCALING Scaling;
typedef struct SCS_WORK Work;
typedef struct SCS_CONE Cone;
/* defined in linSys.h, can be overriden by user */
typedef struct A_DATA_MATRIX AMatrix;
/* this struct defines the data matrix A */
struct A_DATA_MATRIX {
/* A is supplied in column compressed format */
scs_float * x; /* A values, size: NNZ A */
scs_int * i; /* A row index, size: NNZ A */
scs_int * p; /* A column pointer, size: n+1 */
scs_int m, n; /* m rows, n cols */
};
/* struct containing problem data */
struct SCS_PROBLEM_DATA {
/* these cannot change for multiple runs for the same call to scs_init */
scs_int m, n; /* A has m rows, n cols */
AMatrix * A; /* A is supplied in data format specified by linsys solver */
/* these can change for multiple runs for the same call to scs_init */
scs_float * b, *c; /* dense arrays for b (size m), c (size n) */
Settings * stgs; /* contains solver settings specified by user */
};
/* struct containing solver settings */
struct SCS_SETTINGS {
/* settings parameters: default suggested input */
/* these *cannot* change for multiple runs with the same call to scs_init */
scs_int normalize; /* boolean, heuristic data rescaling: 1 */
scs_float scale; /* if normalized, rescales by this factor: 5 */
scs_float rho_x; /* x equality constraint scaling: 1e-3 */
/* these can change for multiple runs with the same call to scs_init */
scs_int max_iters; /* maximum iterations to take: 2500 */
scs_float eps; /* convergence tolerance: 1e-3 */
scs_float alpha; /* relaxation parameter: 1.8 */
scs_float cg_rate; /* for indirect, tolerance goes down like (1/iter)^cg_rate: 2 */
scs_int verbose; /* boolean, write out progress: 1 */
scs_int warm_start; /* boolean, warm start (put initial guess in Sol struct): 0 */
};
/* contains primal-dual solution arrays */
struct SCS_SOL_VARS {
scs_float * x, *y, *s;
};
/* contains terminating information */
struct SCS_INFO {
scs_int iter; /* number of iterations taken */
char status[32]; /* status string, e.g. 'Solved' */
scs_int statusVal; /* status as scs_int, defined below */
scs_float pobj; /* primal objective */
scs_float dobj; /* dual objective */
scs_float resPri; /* primal equality residual */
scs_float resDual; /* dual equality residual */
scs_float resInfeas;/* infeasibility cert residual */
scs_float resUnbdd; /* unbounded cert residual */
scs_float relGap; /* relative duality gap */
scs_float setupTime;/* time taken for setup phase */
scs_float solveTime;/* time taken for solve phase */
};
/* contains normalization variables */
struct SCS_SCALING {
scs_float *D, *E; /* for normalization */
scs_float meanNormRowA, meanNormColA;
};
/* NB: rows of data matrix A must be specified in this exact order */
struct SCS_CONE {
scs_int f; /* number of linear equality constraints */
scs_int l; /* length of LP cone */
scs_int *q; /* array of second-order cone constraints */
scs_int qsize; /* length of SOC array */
scs_int *s; /* array of SD constraints */
scs_int ssize; /* length of SD array */
scs_int ep; /* number of primal exponential cone triples */
scs_int ed; /* number of dual exponential cone triples */
scs_int psize; /* number of (primal and dual) power cone triples */
scs_float * p; /* array of power cone params, must be \in [-1, 1],
negative values are interpreted as specifying the dual cone */
};
/* SCS returns one of the following integers: (zero never returned) */
#define SCS_INFEASIBLE_INACCURATE (-7)
#define SCS_UNBOUNDED_INACCURATE (-6)
#define SCS_SIGINT (-5)
#define SCS_FAILED (-4)
#define SCS_INDETERMINATE (-3)
#define SCS_INFEASIBLE (-2) /* primal infeasible, dual unbounded */
#define SCS_UNBOUNDED (-1) /* primal unbounded, dual infeasible */
#define SCS_UNFINISHED (0) /* never returned, used as placeholder */
#define SCS_SOLVED (1)
#define SCS_SOLVED_INACCURATE (2)
The types scs_float
and scs_int
can be specified by the user, they default
to double
and int
respectively.
The data matrix A
is specified in column-compressed format and the vectors
b
and c
are specified as dense arrays. The solutions x
(primal), s
(slack), and y
(dual) are returned as dense arrays. Cones are specified as
the struct above, the rows of A
must correspond to the cones in the
exact order as specified by the cone struct (i.e. put linear cones before
second-order cones etc.).
Warm-start
You can warm-start SCS (supply a guess of the solution) by setting warm_start in
Data to 1
and supplying the warm-starts in the Sol struct (x
,y
, and s
). All
inputs must be warm-started if any one is. These
are used to initialize the iterates in scs_solve
.
Re-using matrix factorization
If using the direct version you can factorize the matrix once and solve many
times. Simply call scs_init once, and use scs_solve
many times with the same
workspace, changing the input data b
and c
(and optionally warm-starts) for
each iteration. See run_scs.c for an example.
Using your own linear system solver
To use your own linear system solver simply implement all the methods and the
two structs in include/linSys.h
and plug it in.
Solving SDPs
In order to solve SDPs you must have BLAS and LAPACK installed.
Edit scs.mk
to set USE_LAPACK = 1
and point to the location of these
libraries. Without these you can still solve problems using the other cones.
Running make_scs
in Matlab under the matlab
folder will produce two mex
files, one for each of the direct and indirect solvers.
Remember to include the matlab
directory in your Matlab path if you wish to
use the mex file in your Matlab code. The calling sequence is (for the direct version):
[x,y,s,info] = scs_direct(data,cones,settings)
where data is a struct containing A
, b
, and c
, settings is a struct containing
solver options (see matlab file, can be empty),
and cones is a struct that contains one or more of:
+ f
(num primal zero / dual free cones, i.e. primal equality constraints)
+ l
(num linear cones)
+ q
(array of SOCs sizes)
+ s
(array of SDCs sizes)
+ ep
(num primal exponential cones)
+ ed
(num dual exponential cones)
+ p
(array of primal/dual power params).
Type help scs_direct
at the Matlab prompt to see its documentation.
To install SCS as a python package type:
cd <scs-directory>/python
python setup.py install
You may need sudo
privileges for a global installation. Running SCS requires numpy and
scipy to be installed.
After installing the SCS interface, you import SCS using
import scs
This module provides a single function scs
with the following call signature:
sol = scs(data, cone, [use_indirect=false, verbose=true, normalize=true, max_iters=2500, scale=5, eps=1e-3, cg_rate=2, alpha=1.8, rho_x=1e-3])
Arguments in the square brackets are optional, and default to the values on the right of their respective equals signs.
The argument data
is a python dictionary with three elements A
, b
, and
c
where b
and c
are NUMPY arrays (i.e., matrices with a single column)
and A
is a SCIPY sparse matrix in CSC format; if they are not of the proper
format, SCS will attempt to convert them.
The argument cone
is a dictionary with fields f
, l
, q
, s
, ep
,
ed
, and p
(all of which are optional) corresponding to the supported cone types.
The returned object is a dictionary containing the fields sol['x']
, sol['y']
, sol['s']
, and sol['info']
.
The first three are NUMPY arrays containing the relevant solution. The last field contains a dictionary with solver information.
SCS can be called from Java and Scala via the Java Native Interface (JNI).
To compile the necessary libraries, cd
into the java
directory and type
make
. To test a random cone program type make testproblem
.
To solve a problem create a new instance of ConeProgram
with constructor
public ConeProgram(Data d, Cone k, Settings p, IConeSolver solver);
where the IConeSolver
interface can be one of DirectSolver
or IndirectSolver
.
Then call
ConeProgram.solve();
on your instance of ConeProgram
, which will return an instance of
SolutionWithInfo
, containing the solution and information
about the run.
Usage on MAC:
Using the make
command on MAC will generate .class and dylib files but to use them in JAVA you need to compile them into a single jar file.
1- Download SCS source 2- Run make (This will generate dylib and class files). If you get jni.h/jni_w.h error during make. Look for these files and use step 3. 3- (Optional) Place jni_w.h and jni.h in the same folder and run make. 4- Run make testprogram 5- Put all the class files in a new folder named "scs" 7- Build a jar (https://docs.oracle.com/javase/tutorial/deployment/jar/build.html) 8- Import the jar in eclipse. 9- Link the dynamic library (dylib files) -> Project > Properties > Build Path > Open scs.jar > Native Library > Edit > Select the folder with dylib
See usage instructions here.
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