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2 Formulation & Solution

Formulating the problem

Focus on easily solved forms

Modellers of natural phenomena and engineers formulate optimization problems. They know what classes of problems are easy to solve. So, they often draw on a mental library of functions, composition rules to create, for example a convex program.

Dealing with strict inequality constraints

Replace with non-strict inequalities.

Use equivalent formulations (for convexity?)

You can always replace \(f_0(x)\) or \(f_i(x)\) \ with a \(p(f_i(x))\) if p is monotonically increasing with \(f_i\), and if you can translate between \(\argmin f_0(x)\) and \(\argmin p(y)\) or between \(f_i(x) \leq 0\) and \(p(y) \leq c\).

Some formulations are more easily solved than others. In fact, it can turn a non convex problem to an equivalent convex problem.

Eg: \(\min \norm{x}_2 + t\norm{x}_1 \equiv \min \norm{x}_2^{2} + t\norm{x}_1\).

Also, can use the dual if strong duality holds.

Problem parameters

Not variables

These are different from variables. For example, in \(\min \norm{Ax - b}^{2} + l \norm{x}^{2}\), while \(x\) is a variable, l is a parameter to the optimization problem.

Picking the best parameter

With problem parameters unspecified,\ you have a variety of optimization probems. Then, pick the best problem to solve, ie pick the best parameter.

Qualitative judgement

Set various values for the problem parameters, solve them, and pick the parameter whose solution appears best, from the perspective of the application domain.

Theoretical constraints

Theoretical analysis of the problem domain sometimes connect the parameter values with desirable traits in the solutions.

Specification framework engineering

Specification frameworks abstract away from the \ mathematical details, methodological details (which underlying solver to call), make specification easy. They lead to wide application.

It is a good idea to mimic the way people go about naturally formulating optimization problems for a certain class: eg by drawing on a library of functions known to be convex and rules of composition. Eg: cvx for disciplined convex programming.