Fixing typos

cvxguide.ipynb

@@ -216,7 +216,7 @@
     "\\text{subject to }&a_i^\\text{T}x\\leq b_i,\\ i=1,\\ldots,m\n",
     "\\end{align}\n",
     "where $x$, $c$ and $a_i$ for $i=1,\\ldots,m$ belong to $\\mathbb{R}^n$. In general, there is no analytical solution for a LP\n",
-    "problem. A numerical algorithm is therefore required to solve the problem. The earliest algorithms for solving LP problems were the one developed by Kantorovich in 1940 \\cite{Kantorovich40} and the simplex method proposed by George Dantzig in 1947 \\cite{Dantzig91}. In 1978, the Russian mathematician L. G. Khachian developed a polynomial-time algorithm for solving linear programsthe Russian mathematician L. G. Khachian developed a polynomial-time algorithm for solving LP problems. This algorithm was an interior method, which was later improved by Karmarkar \\cite{Karmarkar84}.\n",
+    "problem. A numerical algorithm is therefore required to solve the problem. The earliest algorithms for solving LP problems were the one developed by Kantorovich in 1940 \\cite{Kantorovich40} and the simplex method proposed by George Dantzig in 1947 \\cite{Dantzig91}. In 1978, the Russian mathematician L. G. Khachian developed a polynomial-time algorithm for solving LP problems. This algorithm was an interior method, which was later improved by Karmarkar \\cite{Karmarkar84}.\n",
     "\n",
     "If some of the entries of $x$ are required to be integers, we have a Mixed Integer Linear Programming (MILP) program. A MILP problem is in general difficult to solve (non-convex and NP-complete). However, in practice, the global optimum can be found for many useful MILP problems.\n",
     "\n",
@@ -519,7 +519,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "In this case the optimzal value of the objective function is unique. However, it can be seen in Figure ref{LPfeas} that any point on the line connecting the two points (-2,1) and (1,-2) including the point (0.49742,-1.49742) can be the optimal solution. Therefore, the LP problem ref{LP2} has infinite optimal solutions. The code, however, returns just one of the optimal solutions."
+    "In this case the optimal value of the objective function is unique. However, it can be seen in Figure ref{LPfeas} that any point on the line connecting the two points (-2,1) and (1,-2) including the point (0.49742,-1.49742) can be the optimal solution. Therefore, the LP problem ref{LP2} has infinite optimal solutions. The code, however, returns just one of the optimal solutions."
    ]
   },
   {
@@ -1241,7 +1241,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The correspondin point in the triangle is:"
+    "The corresponding point in the triangle is:"
    ]
   },
   {
@@ -1496,7 +1496,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Generalized inequalities can be defined based on propoer cones. Till now we have seen inequalities for real numbers and elementwise inequalities real vectors. The former type of inequality is defined by the propoer cone of nonnegative real numbers. The later type is defined by the proper cone of the nonnegative orthant ($\\mathbb{R}^n_+$) in $\\mathbb{R}^n$. One natural extension of the optimization problems we have seen so far is to define the inequalities by the proper cone of positive semidefinite matrices. For example, consider the following linear optimization problem:\n",
+    "Generalized inequalities can be defined based on proper cones. Till now we have seen inequalities for real numbers and elementwise inequalities real vectors. The former type of inequality is defined by the propoer cone of nonnegative real numbers. The latter type is defined by the proper cone of the nonnegative orthant ($\\mathbb{R}^n_+$) in $\\mathbb{R}^n$. One natural extension of the optimization problems we have seen so far is to define the inequalities by the proper cone of positive semidefinite matrices. For example, consider the following linear optimization problem:\n",
     "\\begin{align}\n",
     "\\text{minimize}  & c^\\text{T}x\\nonumber\\\\\n",
     "\\text{subject to}& Fx+g\\preccurlyeq_{\\mathcal{S}^n_+} 0\\nonumber\\\\\n",
@@ -1544,7 +1544,7 @@
     "\\begin{align}\n",
     "\\text{minimize}& \\lambda_\\max(A(x))\n",
     "\\end{align}\n",
-    "where $A(x)=A_0+x_1A_1+\\cdots+x_nA_n$ where $A_i$'s are symmetric matrices. This problem can be written as the following SDP problem:\n",
+    "where $A(x)=A_0+x_1A_1+\\cdots+x_nA_n$ where $A_i$'s are symmetric matrices. This can be written as the following SDP problem:\n",
     "\\begin{align}\n",
     "\\text{minimize}&t\\nonumber\\\\\n",
     "               & A(x)\\leq tI\n",
@@ -2212,7 +2212,7 @@
     "\\begin{equation}\n",
     "\\dot x = (A+BK)x\n",
     "\\end{equation}\n",
-    "Now, let us write the stability inequalities for the dual system:\n",
+    "Now, let us write stability inequalities for the dual system:\n",
     "\\begin{align}\n",
     "{\\color{red}Q}>0\\nonumber\\\\\n",
     "(A+B{\\color{red}K}){\\color{red}Q}+{\\color{red}Q}(A+B{\\color{red}K})^\\text{T} < 0\n",
@@ -2355,7 +2355,7 @@
     "\\dot x=&Ax+Bw\\nonumber\\\\\n",
     "z=&Cx+Dw\n",
     "\\end{align}\n",
-    "with $x(0)=0$. It assumed that all the eigenvalues of $A$ have negative real values. In the followin, we will review a few special cases of dissipativity."
+    "with $x(0)=0$. It is assumed that all the eigenvalues of $A$ have negative real values. In the followin, we will review a few special cases of dissipativity."
    ]
   },
   {