Examples of Convex Optimization Problems Where Strong Duality Fails

Introduction to Convex Optimization

Convex optimization is a fundamental area in optimization theory, widely used in various applications such as machine learning, control systems, and signal processing. Convex optimization problems are characterized by the convexity of both the objective function and the feasible set, which ensures that any local minimum is also a global minimum. However, even in the realm of convex optimization, certain conditions need to be met for the strong duality to hold, meaning that the optimal value of the primal problem is equal to the optimal value of the dual problem. This article explores specific examples where strong duality fails in convex optimization problems, highlighting the critical conditions like Slater's condition and the duality gap.

Slater's Condition and Strong Duality in Convex Optimization

Strong duality is a powerful result in convex optimization, stating that under certain conditions, the optimal values of the primal and dual problems are equal. One of these conditions is the Slater's condition, which requires the primal problem to have a strictly feasible point. In other words, there must exist an interior point in the feasible region that satisfies all inequality constraints and non-strict inequality constraints with strict inequalities. When strong duality does not hold, it often means that Slater's condition is not satisfied, leading to a non-zero duality gap.

Slater's Condition Fails in Non-Affine Constraints

The following example illustrates a convex optimization problem where Slater's condition fails due to non-affine constraints, leading to the failure of strong duality. Consider the following optimization problem:

[ p^* inf_{xy} left{ e^{-x} : frac{x^2}{y} leq 0 right}]

subject to

[ xy : y geq 0 ]

This problem is convex because the objective function ( e^{-x} ) is convex and the feasible set is convex. The feasible set is defined by ( x 0 ) and ( y geq 0 ). The primal solution is ( p^* 1 ) when ( x 0 ) and ( y geq 0 ).

Lagrangian Dual Problem

The Lagrangian dual problem is formed by introducing Lagrange multipliers. The Lagrangian is given by:

[ L(x, y, lambda) e^{-x} - lambda left( frac{x^2}{y} right) ]

The dual function is the minimum of the Lagrangian with respect to ( x ) and ( y ):

[ g(lambda) inf_{x, y} left{ e^{-x} - lambda left( frac{x^2}{y} right) : y geq 0 right} ]

The Lagrangian dual problem is then:

[ d^* sup_{lambda geq 0} g(lambda) ]

Evaluating the Dual Function

Let's evaluate the dual function ( g(lambda) ) for specific values of ( lambda ). If ( lambda 0 ) or ( lambda eq 0 ), the Lagrangian dual problem can be analyzed as follows: For ( lambda 0 ), the dual function is simply the optimal value of the primal function: For ( lambda eq 0 ), setting ( x L^2 ) and ( y L^3 ) as ( L to infty ) simplifies the expression, showing that the dual value approaches 0 as ( L to infty ). This illustrates that the duality gap is , the optimal dual value ( d^* ) is strictly less than the primal value ( p^* 1 ), indicating that the duality gap is non-zero and strong duality does not hold.

Conclusion

This example highlights the importance of Slater's condition in convex optimization and the consequences of its failure. When a convex optimization problem does not satisfy Slater's condition, or when the constraints are non-affine, strong duality may not hold, leading to a non-zero duality gap. Understanding these conditions is crucial for practitioners and researchers in convex optimization, ensuring the correct interpretation and application of convex optimization techniques in practical scenarios.