examples/worst_case_analysis.ipynb

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     "data": {
      "text/plain": [
       "5x5 Array{Float64,2}:\n",
       " 0.54178     0.45192   0.328892    0.0123621   0.400576\n",
       " 0.45192     0.694173  0.416407   -0.237798    0.295939\n",
       " 0.328892    0.416407  0.496781    0.0266768   0.31054 \n",
       " 0.0123621  -0.237798  0.0266768   0.414028   -0.181312\n",
       " 0.400576    0.295939  0.31054    -0.181312    0.9     "
      ]
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   "source": [
    "# Generate data for worst-case risk analysis.\n",
    "srand(2);\n",
    "n = 5;\n",
    "r = abs(randn(n, 1))/15;\n",
    "Sigma = 0.9 * rand(n, n) - 0.15;\n",
    "Sigma_nom = Sigma' * Sigma;\n",
    "Sigma_nom -= (maximum(Sigma_nom) - 0.9)"
   ]
  },
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     "data": {
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       "5-element Array{Float64,1}:\n",
       " -0.258385\n",
       "  0.53885 \n",
       " -0.241623\n",
       "  0.751627\n",
       "  0.209527"
      ]
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     "execution_count": 2,
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   "source": [
    "# Form and solve portfolio optimization problem.\n",
    "# Here we minimize risk while requiring a 0.1 return.\n",
    "using Convex, SCS\n",
    "set_default_solver(SCSSolver(verbose=0));\n",
    "w = Variable(n);\n",
    "ret = dot(r, w);\n",
    "risk = sum(quad_form(w, Sigma_nom));\n",
    "problem = minimize(risk, [sum(w) == 1, ret >= 0.1, norm(w, 1) <= 2])\n",
    "solve!(problem);\n",
    "wval = vec(evaluate(w))"
   ]
  },
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     "text": [
      "standard deviation = [0.27]\n",
      "worst-case standard deviation = 0.82\n",
      "worst-case Delta = \n",
      "[-0.0 -0.2 0.19 -0.2 -0.2\n",
      " -0.2 0.0 -0.2 0.2 0.2\n",
      " 0.19 -0.2 -0.0 -0.2 -0.2\n",
      " -0.2 0.2 -0.2 0.0 0.2\n",
      " -0.2 0.2 -0.2 0.2 0.0]\n"
     ]
    }
   ],
   "source": [
    "# Form and solve worst-case risk analysis problem.\n",
    "Sigma = Semidefinite(n);\n",
    "Delta = Variable(n, n);\n",
    "risk = sum(quad_form(wval, Sigma));\n",
    "problem = maximize(risk, [Sigma == Sigma_nom + Delta,\n",
    "                    diag(Delta) == 0,\n",
    "                    abs(Delta) <= 0.2,\n",
    "                    Delta == Delta']);\n",
    "solve!(problem);\n",
    "println(\"standard deviation = \", round(sqrt(wval' * Sigma_nom * wval), 2));\n",
    "println(\"worst-case standard deviation = \", round(sqrt(evaluate(risk)), 2));\n",
    "println(\"worst-case Delta = \");\n",
    "println(round(evaluate(Delta), 2));"
   ]
  }
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