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【漫漫科研路\C&C++】CPLEX解SOCP问题

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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：

​ 图1

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：
图2
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

图3
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）：

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#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
 //  End

static void
createmodel(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 variables
	double 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);
}

int
main(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：

图4

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：
图5

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：

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#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
 //  End

static void
createmodel(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 + x2
	model.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 variables
	double 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);
}

int
main(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：
图6

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