A quadratic programming problem seeks to maximize a...

A quadratic programming problem seeks to maximize a quadratric objective function (with terms like $3 x_{1}^{2}$ or $5 x_{1} x_{2}$ ) subject to a set of linear constraints. Give an example of a quadratic program in two variables $x_{1}, x_{2}$ such that the feasible region is nonempty and bounded, and yet none of the vertices of this region optimize the (quadratic) objective.

Key Concepts

Linear Constraints

Linear constraints are equations or inequalities that limit the values the variables can take and define the boundaries of the feasible region. They are 'linear' because they involve variables only to the first power and do not include products or powers of decision variables, resulting in a polyhedral (often convex) feasible set.

Feasible Region

The feasible region is the set of all points that satisfy the given constraints, forming a bounded (closed and limited) subset of the space in the problem. In quadratic programming, despite the curved nature of the objective function, the feasible region is typically defined by linear inequalities which result in a convex polygon or polyhedron.

Quadratic Programming

Quadratic programming is an optimization framework in which the objective function to be optimized is quadratic and the constraints are linear. This setup is common in many applications, such as portfolio optimization and control systems, where the interplay between a quadratic cost function and linear constraints yields a tractable yet rich problem structure.

Quadratic Objective Function

A quadratic objective function includes terms that are quadratic in the decision variables, such as squared terms and cross-product terms. These functions are typically non-linear and can exhibit curvature, which means that the optimal solution may lie in the interior of the feasible region rather than at a constraint boundary.

Interior Optimum

In some quadratic programming problems, especially those with non-linear objective functions, the optimal solution may not occur at a vertex (extreme point) of the feasible region. Instead, due to the curvature of the quadratic function, the optimum can occur at an interior point where the gradient of the objective function is balanced with the constraints, highlighting the importance of considering both boundary and interior solutions.

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