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Lagrange multipliers are a powerful tool in the field of optimization, particularly for dealing with constraints. Consider a scenario where you wish to minimize or maximize a function subject to certain conditions. The Lagrange multiplier technique introduces additional variables, one for each constraint, which represent the sensitivity of the objective function to the constraint.

For example, if you want to minimize a function such as the primal problem given, which is 'minimize x subject to \(x^2 \-1\leq 0\)', you would introduce a Lagrange multiplier \(\lambda\) that would scale the constraint. The Lagrangian function \(L(x, \lambda) = x + \lambda (x^2 - 1)\) combines the objective and the constraint, and finding the saddle points of this function gives us critical information about the optimal solution to the original problem. In essence, the Lagrange multiplier \(\lambda\) tells us how much the objective function would improve by relaxing the constraint.

Slater's condition provides a sufficient criterion for strong duality to hold in convex optimization problems. The condition essentially asserts that if there exists a strictly feasible point that satisfies the constraints, then the duality gap is zero, meaning the optimal values of the primal and dual problems are equal.

In the context of the given exercise, we explored whether Slater's condition was met for various optimization problems. Consider the problem 'minimize x subject to \(x^2 \-1\)'. Here, we have a strictly feasible point (any point x such that \(-1 < x < 1\)), satisfying Slater's condition. This is vital, as it often implies that not only does an optimal solution exist, but that we can also effectively leverage duality to find or bound this solution.

Strong duality refers to the circumstance where the optimal values of a primal optimization problem and its dual counterpart are equal. This condition doesn't always hold, but when it does, it allows for powerful conclusions regarding the solutions of an optimization problem.

Despite Slater's condition being met in the exercise, we discovered that strong duality did not hold for the problem where the dual function became infeasible. Normally, Slater's condition being satisfied would guarantee strong duality, but this discrepancy indicates that there might be a gap or a specific characteristic of the problem that prevents the duality theorem from applying directly.

Duality can be viewed through a geometric lens, providing an intuitive grasp of the concepts involved. Considering both the primal and dual problems as geometric objects, such as sets or curves, we can visualize their relationship and how one 'shadows' the other.

In simpler optimization problems with one or two variables, you can actually sketch graphs to visualize the feasible region, the level curves of the objective function, and how the Lagrange multipliers stretch these level curves to intersect with the feasible region's boundary. Geometrically, the dual problem can be understood as an attempt to bound the primal problem's solution from below (or above, in case of maximization).

Every optimization problem has a primal form, which is the original problem, and a dual problem, derived from the primal by applying the concept of duality. The primal problem seeks to optimize an objective function subject to constraints, while the dual problem aims to find the best bound on the primal problem's optimal value by using Lagrange multipliers. The primal and dual problems are intricately linked by their solutions and by conditions such as Slater's condition.

The problems we solved in the exercise are instances of primal problems; from each, we derived its corresponding dual problem. The relationship between the primal and dual problems can be subtle: in some cases, solving the dual is easier and offers insightful bounds and clues to the structure of the primal solution, as seen in the exercise.