A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization

Answer for Weakness 1 (Originality):

The crucial contribution of our work is maximizing SR under two constraint simultaneously: the m-sparse (cardinality) constraint and the simplex constraint. While there are indeed a bunch of works elaborating SR maximization under various constraints, few of them consider the former two constraints, especially the m-sparse constraint. This setting makes sense because the m-sparse constraint can accurately control the size of selected assets, and the simplex constraint ensures feasibility of the portfolio in practical issues.

To further improve the completeness and contribution of our work, we have added three main components: 1. sufficient conditions for PGA's convergence to a global optimum; 2. analysis of the convergence rate of PGA; 3. validation of PGA's global optimality through simulation experiments. Please refer to 'Author Rebuttal for Global Response' for more details.

Answer for Weakness 2 (Quality):

Our work is already a complete one that elaborates the best theoretical results on the addressed problem. We have already provided the loosest conditions under which local and even global optimality is guaranteed. Based on the existing results in the original manuscript, we have now proven more intuitive sufficient conditions for the convergence of PGA to a globally optimal solution of model (3.7). For a detailed proof, please refer to 1. Sufficient conditions for PGA's convergence to a global optimum in the 'Author Rebuttal for Global Response'. We also conducted a set of simulation experiments to show that the proposed PGA has a high probability (over 72%) of directly converging to a globally optimal solution. For more details of the simulation experiments, please refer to 3. Validation of PGA's global optimality through simulation experiments in the 'Author Rebuttal for Global Response'.

Besides, we have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since this proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding.

We would like to add that the non-convex fractional model with the non-convex $\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.

Answer for Weakness 3 (Significance):

This is because few approaches are proposed to solve the proposed problem (3.3). We have to compare our method with other ones that solve simpler models like (2.6) and (2.8). For another example, [Kim et al., 2016] develops a semidefinite programming (SDP) relaxation of (3.3). As suggested by Reviewer pxYT, this method fails to control cardinality exactly and actually solves another simplified optimization model. We add this method as comparison and the results in Response Table 3 (see the attached PDF file) show that it is not so good as ours.

Answer for Major Questions:

(1) The self-financing constraint restricts the position that an investor can use to ensure feasibility. For example, suppose there are two assets A and B. An investor can eligibly invest 40% and 60% of the whole position on A and B, respectively. The self-financing constraint is (40% + 60%)=1 in this case. But if without a self-financing constraint, the investor may invest 50% and 70% on A and B, respectively. Then the whole position becomes (50%+70%)>1, which is infeasible. The long-only constraint means that an investor can only buy assets instead of short-selling, which is a conventional setting in portfolio optimization.

(2) These are ordinary portfolio optimization models. Some of them are also competitors in the experiments. Since SR maximization originates from ordinary portfolio optimization, we provide a subsection for such background knowledge to the readers.

(3) It means "guarantee the convergence to a locally optimal solution".

(4) Thank you for mentioning the validity of the conditions in item $(i)$ of Theorem 2. Based on this question, we have proven more intuitive sufficient conditions for the convergence of PGA to a globally optimal solution of model (3.7). For a detailed proof, please refer to 1. Sufficient conditions for PGA's convergence to a global optimum in the 'Author Rebuttal for Global Response'.

Specifically, if one of the following two conditions holds for the limit point $v^*$ of the sequence $\\{v^k\\}$ generated by PGA:

(i) $\\|v^*\\|_0<m$,

(ii) $\\|v^*\\|_0=m$ and $\nabla_i f(v^*)>-\epsilon\cdot\min\\{v_i^*|i\in{\rm supp}(v^*)\\}$ for all $i\in\mathbb{N}\backslash{\rm supp}(v^*)$,

then the conditions in item $(i)$ of Theorem 2 holds, that is, $v^*$ is a globally optimal solution of of model (3.7).

(5) We have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since this proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding.

Answer for Minor Questions:

It is a shorting (selling) based strategy that exploits market crash events.

Answer for Suggestions:

We appreciate the reviewer pointing out the issues with citation numbers and grammatical errors in the manuscript. We will thoroughly review the entire manuscript and make the necessary corrections.

Answer for Weakness 1:

In modern portfolio management, it is widely-recognized that the number of selected assets should be restricted to a manageable size, in order to keep simplicity and save time and financial costs. Managerial strategies provide an approach to achieve this objective. However, the managerial approaches still require intensive administration and abundant experience in management and finance. Hence researchers turn to sparsity models for solutions via the computational approaches.

Answer for Weakness 2:

Lower or upper bound of portfolio weights, as well as the turnover constraint, can also be deployed in our method, as long as they are convex and have closed-form proximal mapping. One can replace the simplex constraint by the afore-mentioned constraints. The theory and the algorithm throughout the paper still hold.

Answer for Weakness 3:

We have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since the proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding. This convergence rate result implies that the computational complexity of the proposed PGA is $O(TN\varepsilon^{-1})$, where $T$ and $N$ denote the window size and the number of assets, respectively, $\varepsilon$ is the convergence tolerance in the objective function value.

Response Table 2 in the attached PDF shows that the running time of mSSRM-PGA is competitive to those of the competitors. Hence mSSRM-PGA has high computational efficiency besides good investing performance.

Answer for Weakness 4:

This method fails to control cardinality exactly and actually solves another simplified optimization model. We add this method as comparison and the results in Response Table 3 show that it is not so good as ours.

Answer for Questions:

Please note that (3.4) is an equivalent optimization model to (3.3), but other general m-sparse mean-variance optimization models may not be. Solving a general m-sparse mean-variance optimization model may lead to a solution far from the locally optimal ones of (3.3). Therefore, we have carefully deduced (3.4) from (3.3) while preserving equivalence, which is a novel contribution.

Additionally, we have now provided two important theoretical results beyond the original manuscript. First, we established more intuitive sufficient conditions for the proposed PGA algorithm to converge to a global optimal solution. Second, we proved that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. For more details of these two results, please refer to Point 1 and Point 2 in the 'Author Rebuttal for Global Response'. We believe that these new results significantly improve the completeness and contribution of our paper.

We would like to add that the non-convex fractional model with the non-convex $\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.

Answer for Weakness 1

In fact, under certain conditions, our method can directly converge to a globally optimal solution. Based on Theorem 2 $(i)$ in the original manuscript, we have now further proven more intuitive sufficient conditions for convergence to global optimum. For a detailed proof, please refer to 1. Sufficient conditions for PGA's convergence to a global optimum in the 'Author Rebuttal for Global Response'.

Specifically, if one of the following two conditions holds for the limit point $v^*$ of the sequence $\\{v^k\\}$ generated by PGA:

(1) $\\|v^*\\|_0<m$,

(2) $\\|v^*\\|_0=m$ and $\nabla_i f(v^*)>-\epsilon\cdot\min\\{v_i^*|i\in{\rm supp}(v^*)\\}$ for all $i\in\mathbb{N}\backslash{\rm supp}(v^*)$,

then $v^*$ is a globally optimal solution of of model (3.7).

We would like to add that the non-convex fractional model with the non-convex $\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.

To further demonstrate that our proposed method can indeed converge to the global optimum of the model, we conducted additional simulation experiments. For more details of the simulation experiments, please refer to 3. Validation of PGA's global optimality through simulation experiments in the 'Author Rebuttal for Global Response'. In our simulations with 10,000 random data sets for each of the three different initializations, over 72% of the experiments for each initialization showed that both the normalized error of the iterative sequence (NEIS) $\\|v^k-v^*\\|_2/\\|v^*\\|_2$ and the normalized error of the function value (NEFV) $|f(v^k)-f(v^*)|/|f(v^*)|$ obtained after 500 iterations of PGA were less than $10^{-10}$. We show the plots of NEIS and NEFV in Response Figure 1, and show in Response Table 1 these two normalized errors obtained after 500 iterations of PGA in ten simulation experiments (see the attached PDF file).

In 10,000 experiments, we averaged the normalized errors for cases that did not converge to the global optimum ($>10^{-5}$). The average NEIS values for the three different initializations were 0.5132, 0.5550, and 0.6123, while the average NEFV values were 0.0889, 0.1080, and 0.1334, respectively. This indicates that even when not converging to the global optimum, the local optima achieved by our algorithm exhibit good overall performance.

Answer for Weakness 2:

Thanks for the insightful comments regarding convergence rate. We have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since this proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding.

Answer for Question 1:

We have carefully transformed the fractional model (3.3) into an equivalent quadratic model (3.4) in subtractive form. Hence we just need to solve the quadratic model (3.4), which is more robust than directly solving the fractional model (3.3) with respect to the data noise.

Answer for Question 2:

The time complexity of S1, S2 and S3 is $O(TN+Nlog(N))$, and the time complexity of PLCT is $O(TN+N^2)$, where $T$ and $N$ denote the window size and the number of assets, respectively. On the other hand, the time complexity of SSMP, SPOLC, SSPO, IPSRM-D, MAXER and mSSRM-PGA is $O(TN\epsilon^{-1})$, where $\epsilon$ denotes the convergence tolerance in the objective function value. Response Table 2 in the attached PDF shows the running times for these methods, which are consistent with their time complexities. Therefore, mSSRM-PGA is competitive in time complexity, and achieves good investing performance.

In fact, our method guarantees a locally optimal solution to the non-convex optimization in a general case, which is consistent with standard optimization theory. Only under certain conditions (Theorem 2 $(i)$), the locally optimal solution become a globally optimal solution.

Additionally, based on Theorem 2 $(i)$, we have now further proven more intuitive sufficient conditions for global optimum, during the process of rebuttal. For a detailed proof, please refer to 1. Sufficient conditions for PGA's convergence to a global optimum in the 'Author Rebuttal for Global Response'.

Specifically, if one of the following two conditions holds for the limit point $v^*$ of the sequence $\\{v^k\\}$ generated by PGA:

(1) $\\|v^*\\|_0<m$,

(2) $\\|v^*\\|_0=m$ and $\nabla_i f(v^*)>-\epsilon\cdot\min\\{v_i^*|i\in{\rm supp}(v^*)\\}$ for all $i\in\mathbb{N}\backslash{\rm supp}(v^*)$,

then $v^*$ is a globally optimal solution of of model (3.7).

We would like to add that the non-convex fractional model with the non-convex $\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.

To further demonstrate that our proposed method has a high probability of converging to the global optimum of the model, we conducted additional simulation experiments. For more details of the simulation experiments, please refer to 3. Validation of PGA's global optimality through simulation experiments in the 'Author Rebuttal for Global Response'. In our simulations with 10,000 random data sets for each of the three different initializations, over 72% of the experiments for each initialization showed that both the normalized error of the iterative sequence $\\|v^k-v^*\\|_2/\\|v^*\\|_2$ and the normalized error of the function value $|f(v^k)-f(v^*)|/|f(v^*)|$ obtained after 500 iterations of PGA was less than $10^{-10}$ from the global optimum. We show the plots of $\\|v^k-v^*\\|_2/\\|v^*\\|_2$ and $|f(v^k)-f(v^*)|/|f(v^*)|$ in Response Figure 1, and show in Response Table 1 these two normalized errors obtained after 500 iterations of PGA in ten simulation experiments (see the attached PDF file).