IBM CPLEX可以解SOCP问题，但是需要先将这个SOCP问题化为指定的格式。本文首先介绍SOCP问题，然后举例介绍如何将SOCP问题转化为CPLEX认可的输入格式并求解。
SOCP的介绍
关于SOCP问题的介绍，可以参考Boyd等人写的Convex Optimization 或者是维基百科的SOCP词条 。这里摘录Convex Optimization一书中关于SOCP的定义：
CPLEX求解SOCP问题
使用CPLEX求解SOCP问题，一般需要将问题转化为CPLEX可以识别的格式。CPLEX的例子ilosocpex1(位于安装目录的examples文件夹内，例如：C:\Program Files\IBM\ILOG\CPLEX_Enterprise_Server129\CPLEX_Studio\cplex\examples)给予了格式说明：
下面我们首先给一个直接可以使用CPLEX求解的例子，然后在此基础上考虑一个更一般的例子（需要变量替换来符合格式）。
一个简单的例子
如上图所示，q1, q2可以直接转化为前面提到的CPLEX认可格式。下面给出源代码如下（注意项目需要预先配置好，配置请见上一篇博文）：
#include <cmath>#include <string>#include <iostream>#include <ilcplex/ilocplex.h>ILOSTLBEGIN // import namespace std// Tolerance for testing KKT conditions.#define TESTTOL 1e-9// Tolerance for barrier convergence.#define CONVTOL 1e-9// A Simple Example// Minimize// obj: x1 + x2 + x3 + x4 + x5 + x6// Subject To// c1: x1 + x2+ x5= 8// c2: x3 + x5 + x6 = 10// q1: x1 >= |(x2, x3)| ---->>>> q1: [ -x1^2 + x2^2 + x3^2 ] <= 0 and x1 >=0// q2: x4 >= |x5| ---->>>> q2: [ -x4^2 + x5^2 ] <= 0 and x4 >=0// Bounds// x2 Free// x3 Free// x5 Free// x6 Free// Endstatic voidcreatemodel(IloModel& model, IloObjective &obj, IloNumVarArray &x,IloRangeArray &rngs, IloIntArray& cone){IloEnv env = model.getEnv();// Create variables.x.add(IloNumVar(env, 0, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, 0, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));// Create objective function and immediately store it in return value.// obj = IloMinimize(env, x1 + x2 + x3 + x4 + x5 + x6);obj = IloMinimize(env, x[0]+x[1] + x[2] + x[3] + x[4] + x[5]);// Setup model.model.add(x[0] + x[1] + x[4]==8);model.add(x[2] + x[4] + x[5]==10);model.add(-x[0] * x[0] + x[1] * x[1] + x[2] * x[2]<=0);//equal to model.add(-x[3] * x[3] + x[4] * x[4]<=0), useful for lots of variablesdouble a[] = {0,0,0, -1,1 };IloExpr temp(env);for (IloInt i = 3; i < 5; i++){temp += a[i] * x[i] * x[i];}model.add(temp <= 0);temp.end();model.add(obj);}intmain(void){IloEnv env;int retval = -1;try {// Create the model.IloModel model(env);IloCplex cplex(env);IloObjective obj(env);IloNumVarArray vars(env);IloRangeArray rngs(env);IloIntArray cone(env);createmodel(model, obj, vars, rngs, cone);// Extract model.cplex.extract(model);// Solve the problem. If we cannot find an _optimal_ solution then// there is no point in checking the KKT conditions and we throw an// exception.cplex.setParam(IloCplex::Param::Barrier::QCPConvergeTol, CONVTOL);if (!cplex.solve() || cplex.getStatus() != IloAlgorithm::Optimal)throw string("Failed to solve problem to optimality");IloNumArray vals_x(env);env.out() << "Solution status = " << cplex.getStatus() << endl;env.out() << "Solution value = " << cplex.getObjValue() << endl;cplex.getValues(vals_x, vars);env.out() << "Values = " << vals_x << endl;env.end();}catch (IloException &e) {cerr << "IloException: " << e << endl;if (env.getImpl())env.end();::abort();}catch (string& e) {cerr << e << endl;if (env.getImpl())env.end();::abort();}return retval;}
运行结果如下图：
一个更一般的例子
在前面例子的基础上，我们只是改变了约束q1，使其更一般化，如下图所示：
为了将q1转化为合适的格式，我们使用变量替换 $x_7=x_1+x_2 $ 。因此只需在前面源代码中更改 createmodel 函数中的部分代码。为保持代码完整性，我们依旧给出完整的代码：
#include <cmath>#include <string>#include <iostream>#include <ilcplex/ilocplex.h>ILOSTLBEGIN // import namespace std// Tolerance for testing KKT conditions.#define TESTTOL 1e-9// Tolerance for barrier convergence.#define CONVTOL 1e-9// A Simple Example// Minimize// obj: x1 + x2 + x3 + x4 + x5 + x6// Subject To// c1: x1 + x2+ x5= 8// c2: x3 + x5 + x6 = 10// q1: x1 + x2 >= |(x1, x2, x3)| ---->>>> q1:[-x7^2+x1^2+x2^2+x3^2]<=0 and x7=x1+x2>=0// q2: x4 >= |x5| ---->>>> q2: [ -x4^2 + x5^2 ] <= 0 and x4 >=0// Bounds// x1 Free// x2 Free// x3 Free// x5 Free// x6 Free// Endstatic voidcreatemodel(IloModel& model, IloObjective &obj, IloNumVarArray &x,IloRangeArray &rngs, IloIntArray& cone){IloEnv env = model.getEnv();// Create variables.x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, 0, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, -IloInfinity, IloInfinity));x.add(IloNumVar(env, 0, IloInfinity));// x7// Create objective function and immediately store it in return value.// obj = IloMinimize(env, x1 + x2 + x3 + x4 + x5 + x6);obj = IloMinimize(env, x[0]+x[1] + x[2] + x[3] + x[4] + x[5]);// Setup model.model.add(x[0] + x[1] + x[4]==8);model.add(x[2] + x[4] + x[5]==10);model.add(x[6] - x[0] - x[1] == 0); // x7 = x1 + x2model.add(-x[6] * x[6]+ x[0] * x[0] + x[1] * x[1] + x[2] * x[2]<=0); //[-x7^2+x1^2+x2^2+x3^2]<=0//equal to model.add(-x[3] * x[3] + x[4] * x[4]<=0), useful for lots of variablesdouble a[] = {0,0,0, -1,1 };IloExpr temp(env);for (IloInt i = 3; i < 5; i++){temp += a[i] * x[i] * x[i];}model.add(temp <= 0);temp.end();model.add(obj);}intmain(void){IloEnv env;int retval = -1;try {// Create the model.IloModel model(env);IloCplex cplex(env);IloObjective obj(env);IloNumVarArray vars(env);IloRangeArray rngs(env);IloIntArray cone(env);createmodel(model, obj, vars, rngs, cone);// Extract model.cplex.extract(model);// Solve the problem. If we cannot find an _optimal_ solution then// there is no point in checking the KKT conditions and we throw an// exception.cplex.setParam(IloCplex::Param::Barrier::QCPConvergeTol, CONVTOL);if (!cplex.solve() || cplex.getStatus() != IloAlgorithm::Optimal)throw string("Failed to solve problem to optimality");IloNumArray vals_x(env);env.out() << "Solution status = " << cplex.getStatus() << endl;env.out() << "Solution value = " << cplex.getObjValue() << endl;cplex.getValues(vals_x, vars);env.out() << "Values = " << vals_x << endl;env.end();}catch (IloException &e) {cerr << "IloException: " << e << endl;if (env.getImpl())env.end();::abort();}catch (string& e) {cerr << e << endl;if (env.getImpl())env.end();::abort();}return retval;}
运行结果如下图：
