爲什麼solve.QP返回不可行解

Question

我想嘗試在河的均方差最優化示例我用這個簡單的例子開始：

> data(edhec) 
> col.mean <- colMeans(edhec) 
> Dmat <- cov(edhec) 
> Amat <- matrix(1,nrow=nrow(Dmat)) 
> Amat <- cbind(Amat,col.mean) 
> Amat <- cbind(Amat,diag(nrow(Dmat))) 
> bvec <- c(1,0.005,rep(0,nrow(Dmat))) 
> sol <- solve.QP(Dmat = 2*Dmat,dvec = 0.5*col.mean,Amat = Amat,bvec = bvec,meq = 1) 
> sol$solution 
[1] 8.882101e-17 0.000000e+00 6.661437e-01 1.990021e-17 1.900749e-15 -3.685111e-17 2.419712e-16 3.338563e-01 
[9] -3.903081e-16 7.237917e-18 1.093498e-15 0.000000e+00 1.243608e-15 
> 

但是似乎這個解決方案是不可行的。有兩個砝碼其是負頂撞約束之一：

> Amat 
          col.mean       
Convertible Arbitrage 1 0.006408553 1 0 0 0 0 0 0 0 0 0 0 0 0 
CTA Global    1 0.006489474 0 1 0 0 0 0 0 0 0 0 0 0 0 
Distressed Securities 1 0.007953289 0 0 1 0 0 0 0 0 0 0 0 0 0 
Emerging Markets  1 0.008246053 0 0 0 1 0 0 0 0 0 0 0 0 0 
Equity Market Neutral 1 0.006002632 0 0 0 0 1 0 0 0 0 0 0 0 0 
Event Driven   1 0.007622368 0 0 0 0 0 1 0 0 0 0 0 0 0 
Fixed Income Arbitrage 1 0.004230921 0 0 0 0 0 0 1 0 0 0 0 0 0 
Global Macro   1 0.007672368 0 0 0 0 0 0 0 1 0 0 0 0 0 
Long/Short Equity  1 0.007759868 0 0 0 0 0 0 0 0 1 0 0 0 0 
Merger Arbitrage  1 0.006784868 0 0 0 0 0 0 0 0 0 1 0 0 0 
Relative Value   1 0.006701316 0 0 0 0 0 0 0 0 0 0 1 0 0 
Short Selling   1 0.004161184 0 0 0 0 0 0 0 0 0 0 0 1 0 
Funds of Funds   1 0.005918421 0 0 0 0 0 0 0 0 0 0 0 0 1 

的第一列對應於加權和的約束應爲1。第二列是目標返回的約束。其餘的列正好是負數，不包括負值。關於solve.QP的幫助，約束被稱爲A^T b> = b_0。換位Amat看起來

> t(Amat) 
     Convertible Arbitrage CTA Global Distressed Securities Emerging Markets Equity Market Neutral Event Driven 
        1.000000000 1.000000000   1.000000000  1.000000000   1.000000000 1.000000000 
col.mean   0.006408553 0.006489474   0.007953289  0.008246053   0.006002632 0.007622368 
        1.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 1.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   1.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  1.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   1.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 1.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
        0.000000000 0.000000000   0.000000000  0.000000000   0.000000000 0.000000000 
     Fixed Income Arbitrage Global Macro Long/Short Equity Merger Arbitrage Relative Value Short Selling 
        1.000000000 1.000000000  1.000000000  1.000000000 1.000000000 1.000000000 
col.mean   0.004230921 0.007672368  0.007759868  0.006784868 0.006701316 0.004161184 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        1.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 1.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  1.000000000  0.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  1.000000000 0.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 1.000000000 0.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 1.000000000 
        0.000000000 0.000000000  0.000000000  0.000000000 0.000000000 0.000000000 
     Funds of Funds 
      1.000000000 
col.mean 0.005918421 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      0.000000000 
      1.000000000 

和矢量B0似乎也是正確的：

> bvec 
[1] 1.000 0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 
> 

的問題是，爲什麼solve.QP返回一個不可行的解決方案？

Answer 1

+/- 3e-16的值在數值上爲零，在浮點表示的範圍內。在你的解決方案中基本上有兩個非零權重，這兩個都是正數。

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