This paper proposes an optimization method for the Equality Set Projection algorithm to compute the orthogonal projection of polytopes. However, its computational burden significantly increases for the case of dual degeneracy, which limits the application of the algorithm. Two improvements have been proposed to solve this problem for the Equality Set Projection algorithm: first, a new criterion that does not require a discussion of the uniqueness of the solution in linear programming, which simplifies the algorithm process and reduces the computational cost; and second, an improved method that abandons the calculation of a ridge's equality set to reduce the computational burden in the case of high-dimensional dual degeneracy.

Orthogonal projection is valuable for both theoretical research

Traditional projection algorithms can be grouped into three classes, i.e., Fourier elimination

The Equality Set Projection (ESP) algorithm was proposed in

Dual degeneracy is a frequently encountered phenomenon in control applications. For instance, when variables have constant maximum and minimum constraints, the occurrence of dual degeneracy is more likely. Unfortunately, this limitation curtails the wider application of ESP algorithms in the control field. Therefore, the primary motivation behind enhancing the ESP method is to improve its performance under dual-degeneracy conditions. We propose a novel criterion based on the non-zero rows of the null-space matrix to address the issue of discussing the uniqueness of solutions under such conditions. Furthermore, we introduce an improved strategy that eliminates the need to search for equation sets under dual-degeneracy conditions. These two advancements render the ESP algorithm more concise, efficient, and easily implementable. Consequently, the algorithm's performance in the situations involving dual degeneracy has been significantly enhanced, which paves the way for further promotion and application of ESP in engineering practices.

The paper is organized as follows. The preliminary knowledge and the workflow of the original ESP algorithm are introduced in Sect. 2. The two proposed improvement strategies are given in Sects. 3 and 4. Numerical simulation comparisons between the original ESP and the improved ESP are reported in Sect. 5, and final conclusions given in Sect. 6.

A polytope is a bounded polyhedron defined by the intersection of closed half-spaces. Let

The affine hull of every face of the polytope is

If

The following results can be found in

We introduce the workflow of the original ESP algorithm in this subsection. A brief introduction is given here, and the details can be found in

The algorithm is initialized by discovering a random facet

The ESP algorithm consists of three key oracles as given below. Shooting Oracle:

Randomly choose

Get

If the linear programming (1) in Step 1 is not dual-degenerate, i.e., (1) has a unique optimizer, then stop. Otherwise,

For each row

If

Conduct singular value decomposition

Solve the following linear programming:

If LP (2) is not dual-degenerate, i.e., LP (2) has a unique optimizer

Go to Step 4.

For each row

In the original ESP algorithm, expressions (1) and (2) in the SHOOT and ADJ oracles need to judge the uniqueness of the optimizer of linear programming (LP), and different steps are taken according to whether the optimizer is unique or not.

In this section, an illustrative example is given at first, the barriers of the original ESP algorithm in the case of dual degeneracy are listed, and the alternative criteria are proposed.

The linear programming (2) in Step 1 of the ADJ oracle is given for illustration, and the solution of (2) satisfies

Given a polytope

Polytope and its projection onto

Figure 2 shows that the ADJ oracle takes the facet

The ADJ oracle in Example 1.

All the points on the red line segment of

However, most of the LP solvers will only give one optimal solution for the LP problem with multiple optimal solutions. If this solution is mistakenly considered the only optimal solution, this may lead to the iteration cycle and increase the computational burden.

In this case, if

The non-uniqueness of the optimal solution is ignored.

Owing to

The barriers of the original ESP algorithm in the case of dual degeneracy can be listed as follows.

Currently, most of the LP solvers do not have the ability to judge the uniqueness of the solutions of linear programming. Although several achievements have been made on the unique conditions of LP solutions

Since the dual degeneracy is only a necessary but insufficient condition for linear programming to have multiple optimizers

Expression (3) implies that, in the case of

According to Example (1), there may be multiple faces

The discussion on the uniqueness of optimizers of LP is to ensure that the obtained

If there exist zero rows in

Let

According to

Let

Next, we will prove that

Since

Combining Expressions (7) and (8) gives us

From Proposition 2.4,

Necessary part: the necessary part is proven by contradiction.

Suppose that each row in

We can denote

Since

Corollary 3.1 is a direct consequence of Theorem 3.1, and it implies that it is no longer necessary to discuss the uniqueness of the solutions in the SHOOT and ADJ oracles. However, it only takes an arbitrary solution as the unique solution and then gets the

By randomly choosing

If there exist zero rows in

Solve the following linear programming.

In this section, we first explain the obstacles to searching for

In the RDG oracle, if

We note that

To improve the computational efficiency of the algorithm when the optimizer of LP is not unique, the search for

Since

In the ADJ oracle, LP (9) and Eq. (10) can be changed to LP (13) and Eq. (14), respectively.

We can verify that the modified algorithm still gets the correct result, though it also adds certain extra computation cost in the case of

In this section, the effectiveness of the proposed method is verified based on certain typical orthogonal projection cases and applications to

We select three cases for numerical simulation, and two of them are dual-degenerate, though one is not dual-degenerate.

The projection of a three-dimensional

The projection of an

The projection of an

Comparative running times for typical cases.

The running time of the original ESP algorithm increases exponentially with dimensions and the number of half-planes of the input polytopes under the dual-degeneracy cases, whereas the improved ESP algorithm significantly reduces the computational burden in the case of dual degeneracy. The running time is also optimized in the case of non-dual degeneracy, since it is no longer necessary to judge the uniqueness of linear programming solutions in the improved ESP algorithm.

We apply the proposed method for the calculation of

Consider the following linear discrete system.

Then, we define the target set as follows.

Results of the

Moreover, the running times of the original ESP and the improved ESP are shown in Fig. 6. When

Comparative running times for the calculation of the

In this paper, the improved ESP algorithm becomes simpler, faster, and easier to implement in the case of dual degeneracy. This is achieved by the following methods. First, an alternative criterion has been proposed to directly obtain the equality set that maximizes the value of

Software code in this study can be requested from the corresponding author.

Research data in this study can be requested from the corresponding author.

BP and WX proposed improved strategies for the ESP algorithm. WX finished the simulation part. YL participated in the correction of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors acknowledge the financial support of the National Natural Science Foundation of China. The authors would like to thank the editors and the anonymous reviewers for their constructive comments and suggestions that have improved the quality of the paper.

This research has been supported by the National Natural Science Foundation of China (grant no. 62103440).

This paper was edited by Daniel Condurache and reviewed by two anonymous referees.