Linearizing Minimum Holdings Constraints in Portfolio Optimization

Summary

The document formulates a linear optimization problem for portfolio allocations. Its objective maximizes a weighted measure of portfolio spread under a budget constraint and other unspecified limits. A further constraint requires the sum, across assets, of the smaller value between each decision variable and its corresponding current holding to meet a threshold. The decision variables represent market values, while current holdings are treated as fixed market values.

The author asks how to express this minimum-based constraint in a linear model, but the document contains no proposed formulation or answer. It therefore gives a useful example of an optimization modeling question, while leaving unresolved whether additional assumptions or variables are needed to represent the intended holdings overlap. No results or tests are reported.

Key ideas

• The decision variables represent proposed portfolio market values, and existing holdings are fixed inputs.
• The constraint sums the smaller of each proposed value and current holding, then compares that total with a threshold.
• The objective maximizes a weighted portfolio measure subject to a budget and other constraints.
• The document poses the linearization question but does not provide a solution.

Tags

# how can I linearize a constraint of the form sum(min(x(i),y(i))) for a linear optimisation problem?


# how can I linearize a constraint of the form sum(min(x(i),y(i))) for a linear optimisation problem?












I have an linear optimisation problem with the objective :

$ max PortfolioSpread(x_1,x_2,....x_N) = ∑_{i=0}^N(x_i*s_i)/budget$

s.t.

- $∑_{i=0}^N x_i = budget$ (+ other constraints)

- $∑_{i=0}^N min⁡(x_i,y_i)≥C$

where decision variables $x_i$ represent market values, $y_i$ represent current holdings expressed in market values while C is the constraint value.

how can I linearize the second constraint?

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