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<p>Although the mainstream literature resorts to numerical iterations for an optimization task, the research path toward exact solutions has recently become clearer; for example, the first, exact, and complete solution to quadratic programming (basis of optimization) has been reported. This paper follows the path and focuses on Problems (1) and (2). (1) We analyze the consistency between the two methodologies to the core; specifically, the analysis clarifies the common ground that supports both Numerical and Exact Optimization. The common ground is equality-constrained quadratic programming. Hence, the theoretical results help to revisit classical problems in, for instance, financial economics. (2) We selectively demonstrate the benchmark problem of Portfolio Selection and formulate the closed-form solution corresponding to positive semidefinite covariance matrices in general, which aligns with Markowitz’s vision and the diversification principle in investment. Financial insights highlight the generalization of Minimum-Variance Set (resp., Efficient Frontier), the geometric structure of which is consistently and normally a parabola (resp., the upper half of a parabola) with respect to the general risk measure. &#xa0;Noting the ubiquitous nature of optimization, various communities have been calling for theoretical support of Problem (1), such as consistency analysis and comparison study. In particular, the operations research community describes solving the problems “the start of a substantial study”. The contributions of this paper are: (i) solution to Problem (1) and its byproduct that helps in analyzing (2); and (ii) unified, closed-form parameterization of Minimum-Variance Set/Efficient Frontier – which theoretically complements the literature gap associated with singular covariance matrices – and its computational enhancement that enriches practical potential. Validations and evaluations are in terms of pedagogical and real-world examples, as well as numerical and hardware experiments on basic computing platforms such as microprocessor and field-programmable gate array. The practical results further strengthen the commercial potential by the computational enhancement (in time and accuracy) and low implementation cost. Overall, the proposed results achieve the research frontier and advance the field.</p>

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The common ground of Numerical Optimization and Exact Optimization: Quadratic programming theory and its financial insights for Portfolio Selection

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月18日

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  • Li-Gang Lin

摘要

Although the mainstream literature resorts to numerical iterations for an optimization task, the research path toward exact solutions has recently become clearer; for example, the first, exact, and complete solution to quadratic programming (basis of optimization) has been reported. This paper follows the path and focuses on Problems (1) and (2). (1) We analyze the consistency between the two methodologies to the core; specifically, the analysis clarifies the common ground that supports both Numerical and Exact Optimization. The common ground is equality-constrained quadratic programming. Hence, the theoretical results help to revisit classical problems in, for instance, financial economics. (2) We selectively demonstrate the benchmark problem of Portfolio Selection and formulate the closed-form solution corresponding to positive semidefinite covariance matrices in general, which aligns with Markowitz’s vision and the diversification principle in investment. Financial insights highlight the generalization of Minimum-Variance Set (resp., Efficient Frontier), the geometric structure of which is consistently and normally a parabola (resp., the upper half of a parabola) with respect to the general risk measure.  Noting the ubiquitous nature of optimization, various communities have been calling for theoretical support of Problem (1), such as consistency analysis and comparison study. In particular, the operations research community describes solving the problems “the start of a substantial study”. The contributions of this paper are: (i) solution to Problem (1) and its byproduct that helps in analyzing (2); and (ii) unified, closed-form parameterization of Minimum-Variance Set/Efficient Frontier – which theoretically complements the literature gap associated with singular covariance matrices – and its computational enhancement that enriches practical potential. Validations and evaluations are in terms of pedagogical and real-world examples, as well as numerical and hardware experiments on basic computing platforms such as microprocessor and field-programmable gate array. The practical results further strengthen the commercial potential by the computational enhancement (in time and accuracy) and low implementation cost. Overall, the proposed results achieve the research frontier and advance the field.