To analyze whether the problem is convex and to find the Lagrange dual, let's proceed systematically.

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1. Convexity Analysis

The problem is:

$\text{Minimize } f(x) = x_1^2 + 2x_2^2 + 2x_1x_2 + x_1 - x_2 - x_3$ Subject to:

1. $x_1 + x_2 + x_3 \leq 1$
2. $x_3 \leq 1$

Objective Function Analysis

The Hessian of the objective function is:

2 & 1 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ The eigenvalues of this Hessian matrix are computed to check if the matrix is positive semidefinite. If all eigenvalues are non-negative, $$f(x)$$ is convex. Calculating the eigenvalues: - Eigenvalues of $$H$$: $$0, 2, 4$$ Since all eigenvalues are non-negative, $$H$$ is positive semidefinite, and the objective function $$f(x)$$ is **convex**. #### Constraint Analysis The constraints $$x_1 + x_2 + x_3 \leq 1$$ and $$x_3 \leq 1$$ are affine (linear) functions, which are both convex. #### Conclusion on Convexity Since the objective function is convex and the constraints are convex, the given problem is a **convex optimization problem**. --- ### **2. Lagrange Dual** To form the Lagrange dual, introduce Lagrange multipliers: - $$\lambda_1 \geq 0$$ for the inequality constraint $$x_1 + x_2 + x_3 \leq 1$$ - $$\lambda_2 \geq 0$$ for the inequality constraint $$x_3 \leq 1$$ #### Lagrangian Function The Lagrangian is: $$L(x_1, x_2, x_3, \lambda_1, \lambda_2) = x_1^2 + 2x_2^2 + 2x_1x_2 + x_1 - x_2 - x_3 + \lambda_1(1 - x_1 - x_2 - x_3) + \lambda_2(1 - x_3)$$ Simplifying: $$L(x_1, x_2, x_3, \lambda_1, \lambda_2) = x_1^2 + 2x_2^2 + 2x_1x_2 + x_1 - x_2 - x_3 + \lambda_1 - \lambda_1x_1 - \lambda_1x_2 - \lambda_1x_3 + \lambda_2 - \lambda_2x_3$$ $$L(x_1, x_2, x_3, \lambda_1, \lambda_2) = x_1^2 + 2x_2^2 + 2x_1x_2 + (1 - \lambda_1)x_1 - (1 + \lambda_1)x_2 - (1 + \lambda_1 + \lambda_2)x_3 + \lambda_1 + \lambda_2$$ #### Dual Function The dual function $$g(\lambda_1, \lambda_2)$$ is obtained by minimizing $$L(x_1, x_2, x_3, \lambda_1, \lambda_2)$$ with respect to $$x_1, x_2, x_3$$. 1. Differentiate $$L$$ w.r.t. $$x_1$$, $$x_2$$, and $$x_3$$: - $$\frac{\partial L}{\partial x_1} = 2x_1 + 2x_2 + (1 - \lambda_1) = 0$$ - $$\frac{\partial L}{\partial x_2} = 4x_2 + 2x_1 - (1 + \lambda_1) = 0$$ - $$\frac{\partial L}{\partial x_3} = -(1 + \lambda_1 + \lambda_2) = 0$$ 2. Solve these equations: - From the third equation, $$(1 + \lambda_1 + \lambda_2) = 0$$, implying $$\lambda_1 + \lambda_2 = -1$$, which is infeasible since $$\lambda_1, \lambda_2 \geq 0$$. Thus, no feasible $$(\lambda_1, \lambda_2)$$ exists that satisfies this. --- ### **Conclusion** - **Convexity**: The problem is convex. - **Lagrange Dual**: The dual formulation appears to have no feasible solution due to the constraint inconsistencies in the Lagrangian minimization step. Adjusting the problem setup or further analysis might be needed. Would you like to explore this dual in more depth or resolve potential inconsistencies? --- ### 5 Related Questions 1. What is the significance of convexity in optimization problems? 2. How do eigenvalues of the Hessian matrix relate to convexity? 3. What are the benefits of solving a problem via the Lagrange dual? 4. How do you interpret the dual variables in optimization problems? 5. Can the problem be reformulated if the dual has no feasible solution? **Tip**: When dealing with convex problems, always verify constraints for consistency before proceeding to dual analysis.