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   "execution_count": 1,
   "id": "e899db8b-20a2-49b7-a8e8-74361b9d5ce8",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import cvxpy as cp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "id": "cee5b993-b83f-46dc-a498-0f7af5668ba6",
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = cp.Variable() # the letter coming after cp. should be capitilized\n",
    "x2 = cp.Variable()\n",
    "objective = cp.Maximize(140*x1 + 235*x2) \n",
    "constraints = [x1 >= 0, x2 >= 0, x1 + x2<=180, x1+2*x2 <= 240, .3*x1 + .1*x2 <=30] # constraints should be defined in an array\n",
    "#constraints+=[x1**2 + 2*x2**2 <=1]  # Can add a convex quadratic constraint (but not a concave one!) += adds a constraint to the array\n",
    "problem = cp.Problem(objective, constraints) # A problem includes an objective (with min/max specified) and a list of constraints\n",
    "problem.solve() #This solves the optimization problem\n",
    "x1_val = x1.value #x1 is still a cvxpy object after solving, but we can save its value like this\n",
    "x2_val = x2.value\n",
    "optimal_val = problem.value #saves optimal value\n",
    "print(x1_val) #returns the value of x1\n",
    "print(x2_val)\n",
    "print(optimal_val)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "aba36a5b-506c-47bf-96a1-222199a51689",
   "metadata": {},
   "outputs": [],
   "source": [
    "#same problem as above\n",
    "x1 = cp.Variable()\n",
    "x2 = cp.Variable()\n",
    "objective_fn = 140*x1 \n",
    "objective_fn += 235*x2 #one can also define an objective function in multiple lines\n",
    "objective = cp.Maximize(objective_fn) \n",
    "constraints = [x1 >= 0, x2 >= 0, x1 + x2<=180, x1+2*x2 <= 240, .3*x1 + .1*x2 <=30]\n",
    "problem = cp.Problem(objective, constraints)\n",
    "problem.solve(verbose = True) #If you want more information about the solving, you can add \"verbose = True\" to the argument of problem.solve.\n",
    "x1_val = x1.value\n",
    "x2_val = x2.value\n",
    "optimal_val = problem.value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "id": "ceced565-a16e-4f17-9b4d-3fec17738016",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Least Squares Problem\n",
    "m=16\n",
    "n=8\n",
    "A = np.random.normal(0,1,[m,n])\n",
    "b = np.random.normal(0,1,[m,1])\n",
    "x = cp.Variable(n)\n",
    "objective = cp.Minimize(cp.sum_squares(A@x - b)) #cvxpy has many built-in functions that you can use to define constraints/objective\n",
    "# objective = cp.Minimize( cp.norm(A@x-b) )  #norm of an affine function - can also try other norms\n",
    "# objective = cp.Maximize( cp.norm(A@x-b) )  # will produce error- input to maximize is not concave\n",
    "problem = cp.Problem(objective)\n",
    "problem.solve()\n",
    "x_val = x.value\n",
    "optimal_val = problem.value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "id": "740f39dd-dc1d-4398-9121-20c89d704478",
   "metadata": {},
   "outputs": [],
   "source": [
    "m=16\n",
    "n=8\n",
    "p=4\n",
    "A = np.random.normal(0,1,[m,n])\n",
    "b = np.random.normal(0,1,[m,1])\n",
    "C = np.random.normal(0,1,[p,n])\n",
    "d = np.random.normal(0,1,[p,1])\n",
    "x = cp.Variable(n)\n",
    "objective = cp.Minimize(cp.norm( A@x -b, 2))\n",
    "constraints = [C@x == d, x >= .1, cp.norm(x,\"inf\") <= 100] # equality constraints use ==\n",
    "problem = cp.Problem(objective,constraints)\n",
    "problem.solve()\n",
    "x_val = x.value\n",
    "optimal_val = problem.value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "id": "fb2fa4e2-c7ce-4ef1-8cda-8e40fa827575",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Infeasible / unbounded problem\n",
    "x1 = cp.Variable()\n",
    "x2 = cp.Variable()\n",
    "constraints = [x1+x2 <= -5] \n",
    "#constraints = [x1+x2 < -5] # Strict inequalities cause an error\n",
    "constraints+=[x1**2 + x2**2 <= 1] # Removing this constraint will make the problem unbounded instead of infeasible.\n",
    "problem = cp.Problem(cp.Minimize(x1+x2),constraints)\n",
    "problem.solve()"
   ]
  }
 ],
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