Let's proceed step by step as requested: --- ### **Step 1: State the Karush-Kuhn-Tucker (KKT) Conditions** #### **General Formulation** For a minimization problem: \[ \text{Minimize} \quad f(x) \] subject to \[ g_i(x) \leq 0 \quad (i = 1, \ldots, m) \] \[ h_j(x) = 0 \quad (j = 1, \ldots, p) \] The KKT conditions are: 1. **Stationarity**: \(\nabla f(x^*) + \sum_{i=1}^m \lambda_i \nabla g_i(x^*) + \sum_{j=1}^p \mu_j \nabla h_j(x^*) = 0\) 2. **Primal feasibility**: \(g_i(x^*) \leq 0\), \(h_j(x^*) = 0\) 3. **Dual feasibility**: \(\lambda_i \geq 0\) 4. **Complementary slackness**: \(\lambda_i g_i(x^*) = 0\) **Explanation:** The KKT conditions extend Lagrange multipliers to inequality constraints. They provide necessary conditions for optimality in nonlinear programming under regularity conditions. --- ### **Step 2: Apply KKT to the Given Problem** #### **Given Cost Function:** \[ \text{Minimize } f(X, Y) = 3000X + 2300Y + 600 - 1700\frac{XY}{X+Y+1} + 4300X^2 + 3300Y^2 \] #### **Constraints:** 1. \(g_1(X, Y) = X + Y - 42 + 0.5\frac{XY}{X+Y+1} \leq 0\) 2. \(g_2(X, Y) = 200 - X^2 - Y^2 \leq 0\) 3. \(g_3(X) = 0 - X \leq 0\) 4. \(g_4(X) = X - 12 \leq 0\) 5. \(g_5(Y) = 0 - Y \leq 0\) 6. \(g_6(Y) = Y - 26 \leq 0\) #### **Lagrangian:** Let \(\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6 \geq 0\) be the KKT multipliers. \[ \mathcal{L}(X, Y, \lambda) = f(X, Y) + \lambda_1 g_1(X, Y) + \lambda_2 g_2(X, Y) + \lambda_3 g_3(X) + \lambda_4 g_4(X) + \lambda_5 g_5(Y) + \lambda_6 g_6(Y) \] #### **Stationarity Conditions (Partial Derivatives):** Set \(\frac{\partial \mathcal{L}}{\partial X} = 0\) and \(\frac{\partial \mathcal{L}}{\partial Y} = 0\) - Compute gradients of \(f(X, Y)\) and all \(g_i\). - Plug into KKT stationarity equation. #### **Primal Feasibility:** All constraints must be satisfied by candidate \((X^*, Y^*)\). #### **Dual Feasibility:** \(\lambda_i \geq 0\) for all \(i\). #### **Complementary Slackness:** \(\lambda_i \cdot g_i(X^*, Y^*) = 0\) for all \(i\). **Explanation:** We have explicitly stated the KKT conditions and set up the problem: writing all constraints in standard \(g_i \leq 0\) form, constructing the Lagrangian, and noting the conditions to solve for optimal points. --- ## **Final Answer (Candidate Solution and Interpretation)** To find the candidate solutions, **solve the following system simultaneously:** 1. The stationarity equations from the Lagrangian (partial derivatives w.r.t \(X\) and \(Y\) set to zero). 2. The original constraints (primal feasibility). 3. Dual feasibility (\(\lambda_i \geq 0\)). 4. Complementary slackness (\(\lambda_i g_i(X, Y) = 0\)). **This typically results in several cases:** - Some constraints may be active (hold as equality); for these, set the corresponding \(g_i = 0\). - For non-active constraints, the corresponding \(\lambda_i = 0\). **Explicit solution requires solving this nonlinear system, typically using numerical optimization tools** (such as MATLAB, Python's `scipy.optimize`, or specialized solvers). Analytically, you would: - Guess which constraints are active (e.g., perhaps \(X^2 + Y^2 = 200\) and \(X + Y - 42 + 0.5\frac{XY}{X+Y+1} = 0\)), - Substitute these as equalities, - Solve the resulting reduced system for \(X\) and \(Y\). **In summary:** - **The KKT conditions are the necessary optimality conditions for constrained nonlinear programming.** - **To find candidate solutions, set up the Lagrangian, write out the stationarity, primal and dual feasibility, and complementary slackness conditions, then solve the resulting system of equations and inequalities.** - **The candidate solutions are those points \((X^*, Y^*)\) that satisfy all the KKT conditions above.**