Methods and systems for analytical-based multifactor multiobjective portfolio risk optimization
First Claim
1. A method for determining optimal solutions along an efficient frontier comprising, a collection of security allocations for a portfolio by using a sequential linear programming technique for a problem having a nonlinear constraint, the method comprising:
- (a) generating a first optimal solution point along the efficient frontier by solving a relaxed linear programming problem with a plurality of linear constraints that excludes the nonlinear constraint using a processor of a computing device, the plurality of linear constraints forming a feasible region;
(b) calculating a first tangent plane for the nonlinear constraint at said first optimal solution point using the processor of the computing device;
(c) shifting the first tangent plane for the nonlinear constraint by a small step size to produce an additional linear constraint using the processor of the computing device, wherein the additional linear constraint lies within the feasible region;
(d) generating a second optimal solution point along the efficient frontier by solving the relaxed linear programming problem with the additional linear constraint using the processor of the computing device;
(e) calculating a second tangent plane for the nonlinear constraint at said second optimal solution point using the processor of the computing device;
(f) shifting the second tangent plane for the nonlinear constraint by a small step size to produce a second additional linear constraint using the processor of the computing device, wherein the second additional linear constraint lies within the feasible region; and
(g) generating additional optimal solution points along the efficient frontier by solving the relaxed linear programming problem with additional linear constraints using the processor of the computing device.
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Accused Products
Abstract
The invention provides systems and methods for determining an efficient frontier, which comprises a collection of security allocations in a portfolio, with multiple, conflicting objectives in a multi-factor portfolio problem. The method includes providing a mathematical model of a relaxation of a problem; generating a sequence of additional constraints; and sequentially applying respective nonlinear risk functions to generate respective adjusted maximum return solutions to obtain an efficient frontier.
140 Citations
20 Claims
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1. A method for determining optimal solutions along an efficient frontier comprising, a collection of security allocations for a portfolio by using a sequential linear programming technique for a problem having a nonlinear constraint, the method comprising:
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(a) generating a first optimal solution point along the efficient frontier by solving a relaxed linear programming problem with a plurality of linear constraints that excludes the nonlinear constraint using a processor of a computing device, the plurality of linear constraints forming a feasible region; (b) calculating a first tangent plane for the nonlinear constraint at said first optimal solution point using the processor of the computing device; (c) shifting the first tangent plane for the nonlinear constraint by a small step size to produce an additional linear constraint using the processor of the computing device, wherein the additional linear constraint lies within the feasible region; (d) generating a second optimal solution point along the efficient frontier by solving the relaxed linear programming problem with the additional linear constraint using the processor of the computing device; (e) calculating a second tangent plane for the nonlinear constraint at said second optimal solution point using the processor of the computing device; (f) shifting the second tangent plane for the nonlinear constraint by a small step size to produce a second additional linear constraint using the processor of the computing device, wherein the second additional linear constraint lies within the feasible region; and (g) generating additional optimal solution points along the efficient frontier by solving the relaxed linear programming problem with additional linear constraints using the processor of the computing device. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 19)
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10. An article of manufacture for determining an efficient frontier comprising a collection of security allocations in a portfolio by using a sequential linear programming technique for a problem having a nonlinear constraint, the article of manufacture comprising:
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a computer readable medium tangibly embodying a computer program executable by a processor of a computer, wherein the computer program contains a model portion that generates a first optimal solution point along the efficient frontier by solving a relaxed linear programming problem with a plurality of linear constraints that form a feasible region, wherein the solving excludes the nonlinear constraint, calculates a first tangent plane for the nonlinear constraint at said first optimal solution point, shifts the first tangent plane for the nonlinear constraint by a small step size to produce an additional linear constraint that lies within the feasible region, and generates a second optimal solution point along the efficient frontier by solving the relaxed linear programming problem with the additional linear constraint; a linear constraint generation portion that generates a sequence of additional linear constraints, wherein said additional linear constraints are an approximation of the nonlinear constraint; and a solution generation portion that sequentially applies the sequence of additional linear constraints to generate additional optimal solution points along the efficient frontier. - View Dependent Claims (11, 12, 13, 14, 15, 16)
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17. An article of manufacture readable by a machine, tangibly embodying a program of instructions executable by the machine to perform method steps for determining an efficient frontier, comprising a collection of security allocations in a portfolio, said method steps comprising:
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(a) generating a first optimal solution point along the efficient frontier by solving a relaxed linear programming problem with a plurality of linear constraints that excludes the nonlinear constraint using a processor of a computing device, the plurality of linear constraints forming a feasible region; (b) calculating a first tangent plane for the nonlinear constraint at said first optimal solution point using the processor of the computing device; (c) shifting the first tangent plane for the nonlinear constraint by a small step size to produce an additional linear constraint using the processor of the computing device, wherein the additional linear constraint lies within the feasible region; (d) generating a second optimal solution point along the efficient frontier by solving the relaxed linear programming problem with the additional linear constraint using the processor of the computing device; (e) calculating a second tangent plane for the nonlinear constraint at said second optimal solution point using the processor of the computing device; (f) shifting the second tangent plane for the nonlinear constraint by a small step size to produce a second additional linear constraint using the processor of the computing device, wherein the second additional linear constraint lies within the feasible region; and (g) generating additional optimal solution points along the efficient frontier by solving the relaxed linear programming problem with additional linear constraints using the processor of the computing device. - View Dependent Claims (18, 20)
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Specification