Residual-driven adaptive momentum average block katzmarz solution method and system

Abstract

The present application relates to the technical field of numerical linear algebra and scientific computing, in particular to a residual-driven adaptive momentum average block Karmarkar method and system. The method steps include: inputting parameters and matrices; initializing the linear feasibility problem estimation solution and inequality, equality part residual; in each iteration, through the residual-driven switching mechanism, when t, the row index set is selected from the inequality row profile with a probability, and when t, the row index set is selected from the equality row profile with a probability; the selected row index is given a convex combination weight; the optimal step size and the optimal momentum coefficient are calculated; the new iteration point combined with the momentum term is generated and the residual is updated; the iteration is repeated until the error limit is met, and the estimation solution is output. Through the residual-driven block switching mechanism, the optimal step size and the optimal momentum parameter setting, the convergence speed and the calculation efficiency of solving large-scale linear feasibility problems are significantly improved.

Description

Technical Field

[0001] This invention relates to the fields of numerical linear algebra and scientific computing, specifically to a residual-driven adaptive momentum-averaged block Katzmalz solution method and system. Background Technology

[0002] Many scientific and engineering computational problems require solving linear feasibility problems. That is, finding vectors Simultaneously satisfying a set of linear equations and a set of linear inequalities ,in The coefficient matrix, For the observation vector, Representing the set of real numbers On A dimensional vector space whose elements are all OK A column of real matrices, Representing the set of real numbers On 3D vector space. This type of problem is widely found in applications such as image reconstruction, signal processing, sparse optimization, machine learning, and distributed computing.

[0003] The Kaczmarz algorithm was originally proposed for solving linear equation systems. The basic iterative format involves projecting the current iteration point sequentially onto the hyperplane defined by each equation. Extending this idea to the linear feasibility problem, the core lies in classifying the projection operations: for rows of equality constraints, the orthogonal projection of the current point to the corresponding hyperplane is directly calculated and updated; for rows of inequality constraints, it is first checked whether the current point already satisfies the inequality—if it does, it is skipped; if not, it is projected onto the corresponding equality boundary hyperplane, i.e., projection correction is only performed on "violated" inequalities. By performing this "selective" projection on violated constraints row by row or block by block, the algorithm gradually minimizes the degree of violation of all equality and inequality constraints, eventually converging to a point within the feasible region. To overcome the sensitivity of classical methods to the order of row processing, the randomized Kaczmarz algorithm and its block variant introduce a row norm-weighted random row selection strategy, allowing each iteration to select working rows or blocks probabilistically, thus achieving an exponential convergence rate that can be explicitly estimated by matrix condition numbers.

[0004] However, existing stochastic Kaczmarz-like algorithms still have significant shortcomings when handling mixed equality and inequality constraints. On the one hand, even when inequalities are transformed into truncated projections, the convergence process remains slow, especially when the degree of violation of inequality and equality constraints is unbalanced, making it difficult for fixed strategies to quickly correct the dominant error. On the other hand, existing methods generally lack effective utilization of historical iteration information, leading to oscillations in the iteration direction near extreme points, affecting global convergence efficiency. In recent years, momentum acceleration techniques have been introduced into the Kaczmarz framework, achieving some acceleration by mixing the current gradient projection direction with the previous search direction. However, how to adaptively select inequality or equality working blocks based on the violation characteristics of the current solution, introduce momentum terms, and simultaneously provide matching optimal step size and optimal momentum coefficients remains an open problem. Therefore, there is an urgent need for an efficient linear feasibility solution method that can dynamically balance different constraint types and integrate adaptive step size and momentum acceleration. Summary of the Invention

[0005] In order to overcome the defects and shortcomings of the existing technology, the present invention provides a residual-driven adaptive momentum-averaged block Katzmalz solution method and system.

[0006] According to a first aspect of the present invention, a residual-driven adaptive momentum-averaged block Katzmalz solution method is provided, comprising the following steps:

[0007] Input steps: Input coefficient matrix Observation vector Initial iteration point Residual ratio Huffman estimation parameters Error limit Inequality line decomposition and line decomposition of equations ;

[0008] Initialization steps: Initialize the solution to the linear feasibility problem to obtain the first iteration point. Calculate the residuals of the inequality part. Residuals in the equation ;

[0009] Iteration steps: Set the iteration stopping condition Execute the first iteration ; Defined as , independent variable , For the set of real numbers superior 3D vector space, To obtain and The largest one, and These are the index sets of the inequality rows and the index sets of the equality rows, respectively. for -norm, Coefficient matrix With the new iteration point Multiply; if ,from Selecting the index set based on probability ,like ,from Selecting the index set based on probability For those already selected Indicators within Assign weights to convex combinations ; Calculate the optimal step size and optimal momentum coefficient ;

[0010] Define the correction direction , For matrix transpose, For matrix of A matrix composed of rows, , For A quasi-diagonal matrix composed of diagonal elements. for The square of, For matrix The OK, for With iteration point Multiply, To obtain column vectors Subscript belongs to The elements of the vector, arranged in their original order, form a sub-column vector, which is obtained by the formula... calculate ;based on renew and The iteration terminates when the stopping condition is met; otherwise, the process from selecting and updating the index set is repeated. and ;

[0011] Output steps: Output the estimated solution to the linear feasibility problem.

[0012] According to a second aspect of the present invention, a residual-driven adaptive momentum-averaged block Katzmalz solution system is provided, comprising the following modules:

[0013] Input module, input coefficient matrix Observation vector Initial iteration point Residual ratio Huffman estimation parameters Error limit Inequality line decomposition and line decomposition of equations ;

[0014] The initialization module initializes the estimated solution for the linear feasibility problem and obtains the first iteration point. Calculate the residuals of the inequality part. Residuals in the equation ;

[0015] The iteration module sets the iteration stop condition. Execute the first iteration ; Defined as , independent variable , For the set of real numbers superior 3D vector space, To obtain and The largest one, and These are the index sets of the inequality rows and the index sets of the equality rows, respectively. for -norm, Coefficient matrix With the new iteration point Multiply; if ,from Selecting the index set based on probability ,like ,from Selecting the index set based on probability For those already selected Indicators within Assign weights to convex combinations ; Calculate the optimal step size and optimal momentum coefficient ;

[0016] Define the correction direction , For matrix transpose, For matrix of A matrix composed of rows, , For A quasi-diagonal matrix composed of diagonal elements. for The square of, For matrix The OK, for With iteration point Multiply, To obtain column vectors Subscript belongs to The elements of the vector, arranged in their original order, form a sub-column vector, which is obtained by the formula... calculate ;based on renew and The iteration terminates when the stopping condition is met; otherwise, the process from selecting and updating the index set is repeated. and ;

[0017] The output module outputs the estimated solution to the linear feasibility problem.

[0018] According to a third aspect of the present invention, a computer-readable storage medium is provided storing a computer program that, when executed by a processor, implements a residual-driven adaptive momentum-averaged block Katzmalz solution method.

[0019] According to a fourth aspect of the present invention, a computing device is provided, including a processor and a memory for storing a processor-executable computer program, wherein when the processor executes the computer program stored in the memory, it implements a residual-driven adaptive momentum-averaged block Katzmalz solution method.

[0020] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0021] Existing stochastic Kaczmarz-type algorithms typically select working blocks directly from all rows based on probability, failing to consider the differences in the degree of violation between inequality and equality constraints, and generally lacking effective utilization of historical iteration information. Selecting row blocks solely based on probability easily leads to insufficient updates for sections with high constraint violation rates, while ignoring historical directional information during the iteration process, thus limiting the algorithm's convergence efficiency when dealing with large-scale linear feasibility problems.

[0022] The proposed residual-driven adaptive momentum-averaged block Kaczmarz (AMABK-RS) solution method effectively utilizes the difference between the residuals in the inequality and equality parts, and introduces a momentum term to leverage historical iteration information, while simultaneously providing a matching optimal step size and optimal momentum coefficient. Specifically, in each iteration, the algorithm automatically determines whether to use an inequality constraint block or an equality constraint block through a residual-driven switching mechanism: when the inequality residual is relatively higher, the algorithm prioritizes selecting a working block from the inequality constraints; otherwise, it selects a working block from the equality constraints. Simultaneously, the algorithm introduces a momentum term, weighting the movement direction of the previous iteration with the optimal momentum coefficient and adding it to the update direction of the current iteration.

[0023] Under this residual-driven and momentum-combined mechanism, if the violation of the current constraint type (such as inequalities) is high, the algorithm will automatically prioritize updating that type of constraint, thereby more effectively reducing the dominant residual. If the historical iteration direction is consistent with the current required update direction, the momentum term will accelerate convergence; if not, unnecessary oscillations will be automatically suppressed through the optimal momentum coefficient. Therefore, the AMABK-RS method adopted in this invention can more intelligently balance the update requirements between inequality and equality constraints, while utilizing historical iteration information to smooth the convergence path, thus significantly improving the algorithm's convergence speed and computational efficiency. Furthermore, by jointly calculating the optimal step size and the optimal momentum coefficient, this invention ensures that the algorithm maintains an efficient approximation of the true solution throughout the iteration process, exhibiting good numerical stability and wide applicability. Attached Figure Description

[0024] Figure 1 This is a flowchart of the residual-driven adaptive momentum-averaged block Katz-Malc solution method of the present invention;

[0025] Figure 2 The residual in the embodiments of the present invention With the number of algorithm iterations A diagram illustrating the comparison results;

[0026] Figure 3 The residual in the embodiments of the present invention A diagram showing the comparison results with the algorithm's running time. Detailed Implementation

[0027] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0028] Example

[0029] like Figure 1As shown, this embodiment provides a residual-driven adaptive momentum-averaged block Kaczmarz solution method, which solves large-scale linear feasibility problems based on the adaptive momentum-averaged block Kaczmarz (AMABK-RS) algorithm, and specifically includes the following steps:

[0030] S1: Input coefficient matrix Observation vector Initial iteration point Residual ratio Hoffman parameter estimation Error limit Inequality line decomposition and line decomposition of equations The inequality row decomposition From the inequality row index set The equation row decomposition is obtained by partitioning. From the set of indexes of the equation row The decomposition yields the following results:

[0031] ,satisfy ,and ;

[0032] ,satisfy ,and ;

[0033] in, and Let represent the number of blocks in the index set of inequality rows and the index set of equality rows, respectively. , , Indicates belonging to The intersection of different partitioned sets Indicates belonging to The intersection of different partitioned sets Represents the empty set. Indicates will All partitioned sets Take the union, Indicates will All partitioned sets Take the union;

[0034] S2: Data initialization, including initializing the estimated solution to the linear feasibility problem and denoting it as... and initialization based on The calculated inequality residuals Residuals in the equation ;

[0035] The calculation method is as follows:

[0036] ;

[0037] in, Representation matrix The OK, Representation matrix The transpose of a line It is a randomly selected set of row indicators. Represented by matrix of A matrix composed of rows, Denotes the Frobenius norm. express The square of, Representation matrix The OK With the initial iteration point Multiply, Representing vectors The One element, step size , Representation matrix transpose, Represents the vector All indexes belong to The column vector formed by arranging the elements in their original order. Representation matrix With the initial iteration point Multiply, Representation matrix and Multiply, for -norm, for The square of the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let these represent the row index sets of inequalities and the row index sets of equality, respectively.

[0038] The inequality part of the residual Residuals in the equation based on The calculation is as follows:

[0039] ;

[0040] ;

[0041] in, and Representing the matrix of and A matrix composed of rows, and They represent vectors respectively All indexes belong to and The column vector formed by arranging the elements in their original order. Representation matrix With the first iteration point Multiply, Representation matrix With the first iteration point Multiply, for -norm, above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let these represent the row index sets of inequalities and the row index sets of equality, respectively.

[0042] S3: Set the iteration stopping condition in the AMABK-RS algorithm. In each iteration, the set of row metrics used in this iteration is automatically determined through a residual-driven switching mechanism. Source, i.e., when When, then decompose from the row of inequalities. Selecting the index set based on probability ,when When that happens, the equation is decomposed from the row of equations. Selecting the index set based on probability For the selected row indicator set Inner row indicators Assign weights to convex combinations The optimal step size is calculated. and optimal momentum coefficient ; set the iteration point Along the correction direction synthesized from the selected row weighted residuals opposite direction Move, while incorporating the direction of movement from the previous iteration. , by optimal step size and optimal momentum coefficient Regulate separately and The magnitude of each contribution is summed to generate a new iteration point. and based on Calculate the residuals of the inequality part Residuals in the equation Repeat the process from selecting the index set to updating the inequalities and the residuals in the equations until... The iteration stops, where, For the error limit, the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index sets of inequalities and the row index sets of equality, respectively. express -norm, Represents the coefficient matrix With the new iteration point Multiply, For observation vectors;

[0043] S31: Each time an iteration is performed, the iteration count is incremented. ;

[0044] S32: Automatically determine the set of constraint row indices to be used in this iteration through a residual-driven switching mechanism. Source, when At that time, from Selecting the index set based on probability ,when At that time, from Selecting the index set based on probability .in, Represents the residual ratio. The residuals are the parts of the inequality. For the residuals of the equation, To perform row decomposition of inequalities, To perform row decomposition of the equation;

[0045] S33: Row Partitioning of Inequalities Chinese arbitrary row indicator set Probability of being selected for:

[0046] ;

[0047] Linear decomposition of equations Chinese arbitrary row indicator set Probability of being selected for:

[0048] ;

[0049] in, Denotes the Frobenius norm. express The square of, , , and Representing the matrix of , , and A matrix composed of rows, and Let represent the number of blocks in the index set of inequality rows and the index set of equality rows, respectively. , ;

[0050] S34: For already selected Indicators within Assigned convex combination weights satisfy:

[0051] , ;

[0052] S35: Based on weighted diagonal matrix Square root weighted diagonal matrix The optimal step size is calculated for three cases, taking into account historical movement direction, block residual, and inner product. and optimal momentum parameters :

[0053] like ,but:

[0054] ; ;

[0055] like and ,but:

[0056] ;

[0057] ;

[0058] like and ,but:

[0059] ;

[0060] ;

[0061] in, ; , ; ; ;

[0062] in, Representation matrix transpose, Representation matrix of A matrix composed of rows, Representation matrix The OK, Representation matrix With iteration point Multiply, Represents the vector All indexes belong to A column vector formed by arranging the elements in their original order. and They represent respectively with and A quasi-diagonal matrix composed of diagonal elements. Representation matrix The OK, express -norm, for The square of the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index sets of inequalities and the row index sets of equality, respectively. , and They represent the first , and The iteration point generated in the next iteration. Indicates the inner product. express The square of, This represents the Hoffman estimation parameters;

[0063] S36: New iteration points are then generated as follows. :

[0064] ;

[0065] in, and These represent the optimal step size and optimal momentum parameters, respectively. ; Indicates the first The set of row indicators selected in the next iteration Representation matrix transpose, Represents the selection matrix of A matrix composed of rows, Representation matrix With iteration point Multiply, Indicates A quasi-diagonal matrix composed of diagonal elements. Indicates the set of row indicators The set of convex combination weights assigned to the row indicators within the data. Representation matrix The OK, express -norm, for The square of, Represents the vector All indexes belong to The column vector formed by arranging the elements in their original order. , and They represent the first , and The iteration point generated in the next iteration. Representation matrix With iteration point Multiply, the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let these represent the row index sets of inequalities and the row index sets of equality, respectively.

[0066] S37: Based on Calculate the residuals of the inequality part Residuals in the equation :

[0067] ;

[0068] ;

[0069] in, express -norm, and Representing the matrix of and A matrix composed of rows, and They represent vectors respectively All indexes belong to and The column vector formed by arranging the elements in their original order. Representation matrix With the new iteration point Multiply, Representation matrix With the new iteration point Multiply, the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let these represent the row index sets of inequalities and the row index sets of equality, respectively.

[0070] S38: Check if the iteration termination condition is met:

[0071] ;

[0072] in, For the error limit, the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index sets of inequalities and the row index sets of equality, respectively. express -norm, Represents the coefficient matrix With the new iteration point Multiply, For observation vectors;

[0073] S4: Output the estimated solution to the linear feasibility problem.

[0074] In this embodiment, a coefficient matrix is ​​set. Each element in the vector is independent and identically distributed and follows a standard normal distribution; the right-hand vector The generation method is as follows: First, the index of the inequality row is defined. With the row index of the equation Then, generate a vector in which each element is independent and identically distributed and follows a standard normal distribution. Then generate Finally in of The position of the line indicator is added to The right-hand vector is generated by uniformly distributing random numbers within a given range. The experimental results in this embodiment are on average at... This is an independent experiment.

[0075] like Figure 2 The figure shows the residuals of the AMABK-RS algorithm proposed in this invention and the existing average block Katzmarz algorithm with only step-size adaptation. Compared with the number of algorithm iterations, where The number of rows in the equation is Initial iteration point Zero vector, residual ratio Hoffman parameter estimation Error limit Regarding row index decomposition, the AMABK-RS algorithm divides the inequality row indices and the equality row indices into equal parts. Share and The AMABK-RS algorithm, which uses only step-size adaptive average block Kaczmarz, works similarly. As can be seen from the figure, the AMABK-RS algorithm reduces the relative recovery error more significantly in one iteration compared to the step-size adaptive average block Kaczmarz algorithm, indicating that the AMABK-RS algorithm performs better.

[0076] like Figure 3 As shown, the residuals are displayed. Compared with the algorithm's running time, where The number of rows in the equation is Initial iteration point Zero vector, residual ratio Hoffman parameter estimation Error limit Regarding row index decomposition, the AMABK-RS algorithm divides the inequality row indices and the equality row indices into equal parts. Share and The step-size adaptive average block Kaczmarz algorithm is the same as above. Compared to the existing step-size adaptive average block Kaczmarz algorithm, the AMABK-RS algorithm proposed in this invention can converge to a given error size more quickly and has better algorithm efficiency.

[0077] The present invention also provides an adaptive momentum-averaged block Katz-Malc solution system based on residual driving, comprising: an input module, an initialization module, an iteration module and an output module;

[0078] The input module is used to input a known coefficient matrix. Observation vector Initial iteration point Residual ratio Hoffman parameter estimation Error limit Inequality line decomposition and line decomposition of equations The inequality row decomposition From the inequality row index set The equation row decomposition is obtained by partitioning. From the set of indexes of the equation row The decomposition yields the following results:

[0079] ,satisfy ,and ;

[0080] ,satisfy ,and ;

[0081] in, and Let represent the number of blocks in the index set of inequality rows and the index set of equality rows, respectively. , , Indicates belonging to The intersection of different partitioned sets Indicates belonging to The intersection of different partitioned sets Represents the empty set. Indicates will All partitioned sets Take the union, Indicates will All partitioned sets Take the union.

[0082] The initialization module is used for data initialization, including initializing the estimated solution to the linear feasibility problem and denoting it as... and initialization based on The calculated inequality residuals Residuals in the equation ;

[0083] The calculation method is as follows:

[0084] ;

[0085] in, Representation matrix The OK, Representation matrix The transpose of a line It is a randomly selected set of row indicators. Represented by matrix of A matrix composed of rows, Denotes the Frobenius norm. express The square of, Representation matrix The OK With the initial iteration point Multiply, Representing vectors The One element, step size , Representation matrix transpose, Represents the vector All indexes belong to The column vector formed by arranging the elements in their original order. Representation matrix With the initial iteration point Multiply, Representation matrix and Multiply, for -norm, for The square of the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let these represent the row index sets of inequalities and the row index sets of equality, respectively.

[0086] The inequality part of the residual Residuals in the equation based on The calculation is as follows:

[0087] ;

[0088] ;

[0089] in, and Representing the matrix of and A matrix composed of rows, and They represent vectors respectively All indexes belong to and The column vector formed by arranging the elements in their original order. Representation matrix With the first iteration point Multiply, Representation matrix With the first iteration point Multiply, for -norm, above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index set of inequalities and the row index set of equality, respectively.

[0090] The iteration module is used to perform iterations of the AMABK-RS algorithm. First, it sets the stopping condition for the iteration. Then, in the AMABK-RS algorithm... In each iteration, the set of constraint row indices used in this iteration is automatically determined through a residual-driven switching mechanism. Source, when When, then decompose from the row of inequalities. Selecting the index set based on probability ,when When that happens, the equation is decomposed from the row of equations. Selecting the index set based on probability ,in, Represents the residual ratio. The residuals are the parts of the inequality. For the residuals of the equation, To perform row decomposition of inequalities, This is a row decomposition of the equation.

[0091] Inequality row decomposition Chinese arbitrary row indicator set The probability of being selected is:

[0092] ;

[0093] Linear decomposition of equations Chinese arbitrary row indicator set The probability of being selected is:

[0094] ;

[0095] in, Denotes the Frobenius norm. express The square of, , , and Representing the matrix of , , and A matrix composed of rows, and Let represent the number of blocks in the index set of inequality rows and the index set of equality rows, respectively. , .

[0096] Selected row indicator set Inner row indicators Assigned convex combination weights satisfy:

[0097] , ;

[0098] Subsequently, based on the weighted diagonal matrix Square root weighted diagonal matrix The optimal step size is calculated for three cases, taking into account historical movement direction, block residual, and inner product. and optimal momentum parameters :

[0099] like ,but:

[0100] ; ;

[0101] like and ,but:

[0102] ;

[0103] ;

[0104] like and ,but:

[0105] ;

[0106] ;

[0107] in, ; , ; ; ;

[0108] in, Representation matrix transpose, Representation matrix of A matrix composed of rows, Representation matrix The OK, Representation matrix With iteration point Multiply, Represents the vector All indexes belong to A column vector formed by arranging the elements in their original order. and They represent respectively with and A quasi-diagonal matrix composed of diagonal elements. Representation matrix The OK, express -norm, for The square of the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index sets of inequalities and the row index sets of equality, respectively. , and They represent the first , and The iteration point generated in the next iteration. Indicates the inner product. express The square of, This represents the Hoffman estimation parameters.

[0109] New iteration points are then generated in the following manner. :

[0110] ;

[0111] in, and These represent the optimal step size and optimal momentum parameters, respectively. . Indicates the first The set of row indicators selected in the next iteration Representation matrix transpose, Represents the selection matrix of A matrix composed of rows, Representation matrix With iteration point Multiply, Indicates A quasi-diagonal matrix composed of diagonal elements. Indicates the set of row indicators The set of convex combination weights assigned to the row indicators within the data. Representation matrix The OK, express -norm, for The square of, Represents the vector All indexes belong to The column vector formed by arranging the elements in their original order. , and They represent the first , and The iteration point generated in the next iteration. Representation matrix With iteration point Multiply, the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index set of inequalities and the row index set of equality, respectively.

[0112] Finally, based on Calculate the residuals of the inequality part Residuals in the equation The calculation method is as follows:

[0113] ;

[0114] ;

[0115] in, express -norm, and Representing the matrix of and A matrix composed of rows, and They represent vectors respectively All indexes belong to and The column vector formed by arranging the elements in their original order. Representation matrix With the new iteration point Multiply, Representation matrix With the new iteration point Multiply, the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index set of inequalities and the row index set of equality, respectively.

[0116] Repeat the iteration until the iteration stopping condition is met. The iteration stopping condition is specifically expressed as:

[0117] ;

[0118] in, For the error limit, the above The function is defined as: , For function The independent variable, Representing the set of real numbers On 3D vector space, Indicates taking and The largest one, and Let represent the row index sets of inequalities and the row index sets of equality, respectively. express -norm, Represents the coefficient matrix With the new iteration point Multiply, For observation vectors;

[0119] The output module is used to output the estimated solution to the linear feasibility problem.

[0120] This invention provides a storage medium, which may be a ROM, RAM, disk, optical disk, or other storage medium. The storage medium stores one or more programs. When the programs are executed by a processor, they implement the residual-driven adaptive momentum-averaged block Katz-Malc solution method of this invention.

[0121] This invention provides a computing device, which may be a desktop computer, laptop computer, smartphone, PDA handheld terminal, tablet computer or other terminal device with display function. The computing device includes a processor and a memory. The memory stores one or more programs. When the processor executes the program stored in the memory, it implements the residual-driven adaptive momentum average block Katzmalz solution method of this invention.

[0122] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A residual-driven adaptive momentum-averaged block Katz-Malc solution method, characterized in that, Includes the following steps: Input steps: Input coefficient matrix Observation vector Initial iteration point Residual ratio Huffman estimation parameters Error limit Inequality line decomposition and line decomposition of equations ; Initialization steps: Initialize the solution to the linear feasibility problem to obtain the first iteration point. Calculate the residuals of the inequality part. Residuals in the equation ; Iteration steps: Set the iteration stopping condition Execute the first iteration ; Defined as , independent variable , For the set of real numbers superior 3D vector space, To obtain and The largest one, and These are the index sets of the inequality rows and the index sets of the equality rows, respectively. for -norm, Coefficient matrix With the new iteration point Multiply; if ,from Selecting the index set based on probability ,like ,from Selecting the index set based on probability For those already selected Indicators within Assign weights to convex combinations ; Calculate the optimal step size and optimal momentum coefficient ; Define the correction direction , For matrix transpose, For matrix of A matrix composed of rows, , For A quasi-diagonal matrix composed of diagonal elements. for The square of, For matrix The OK, for With iteration point Multiply; To obtain column vectors Subscript belongs to The elements, arranged in their original order, form a sub-column vector; by the formula calculate ;based on renew and The iteration terminates when the stopping condition is met; otherwise, the process from selecting and updating the index set is repeated. and ; Output steps: Output the estimated solution to the linear feasibility problem.

2. The residual-driven adaptive momentum-averaged block Katz-Malc solution method according to claim 1, characterized in that, The inequality row decomposition From the inequality row index set The equation row decomposition is obtained by partitioning. From the set of indexes of the equation row The decomposition yields the following results: ,satisfy ,and ; ,satisfy ,and ; in, and Let represent the number of blocks in the index set of inequality rows and the index set of equality rows, respectively. , , Indicates belonging to The intersection of different partitioned sets Indicates belonging to The intersection of different partitioned sets Represents the empty set. Indicates will All partitioned sets Take the union, Indicates will All partitioned sets Take the union.

3. The residual-driven adaptive momentum-averaged block Katz-Malc solution method according to claim 1, characterized in that, In the initialization step The calculation method is as follows: ； in, Representation matrix The transpose of a line It is a randomly selected set of row indicators. Represented by matrix of A matrix composed of rows, Describe the Frobenius norm. express The square of, Representation matrix The OK With the initial iteration point Multiply, Representing vectors The One element; step size Defined as: ,in Representation matrix transpose, Represents the vector All indexes belong to The column vector formed by arranging the elements in their original order. Representation matrix With the initial iteration point Multiply, Representation matrix and Multiply; based on Inequality part of the residual Residuals in the equation The calculation is as follows: ； ； in, and Representing the matrix of and A matrix composed of rows, and They represent vectors respectively All indexes belong to and The column vector formed by arranging the elements in their original order. Representation matrix With the first iteration point Multiply, Representation matrix With the first iteration point Multiply.

4. The residual-driven adaptive momentum-averaged block Katz-Malc solution method according to claim 1, characterized in that, Select row indicator set from the corresponding row partitioning The probability rule is: Dissecting the inequality rows Selecting row indicator set probability Its definition is: ; Dissecting the equation lines Selecting row indicator set probability Its definition is: ; in, and Representing the matrix of and A matrix composed of rows, , , and Represent matrices respectively , , and The Frobenius norm.

5. The residual-driven adaptive momentum-averaged block Katz-Malc solution method according to claim 1, characterized in that, The selected Indicators within Assigned convex combination weights satisfy: ,and .

6. The residual-driven adaptive momentum-averaged block Katz-Malc solution method according to claim 1, characterized in that, Based on weighted diagonal matrix Square root weighted diagonal matrix The optimal step size is calculated for three cases, taking into account historical movement direction, block residual, and inner product. and optimal momentum parameters : like ,but: ； ； like and ,but: ； ； like and ,but: ； ； in, ; , ; ; , indicating that A quasi-diagonal matrix composed of diagonal elements; , and They represent the first , and The iteration point generated in the next iteration. Indicates the inner product. express The square of.

7. The residual-driven adaptive momentum-averaged block Katz-Malc solution method according to claim 1, characterized in that, After iteration, based on Inequality part of the residual Residuals in the equation The updated calculation formula is as follows: ； ； in, Representation matrix With the new iteration point Multiply, Representation matrix With the new iteration point Multiply.

8. A residual-driven adaptive momentum-averaged block Katz-Malc solver system, characterized in that, Includes the following modules: Input module, input coefficient matrix Observation vector Initial iteration point Residual ratio Huffman estimation parameters Error limit Inequality line decomposition and line decomposition of equations ; The initialization module initializes the estimated solution for the linear feasibility problem and obtains the first iteration point. Calculate the residuals of the inequality part. Residuals in the equation ; The iteration module sets the iteration stop condition. Execute the first iteration ; Defined as , independent variable , For the set of real numbers superior 3D vector space, To obtain and The largest one, and These are the index sets of the inequality rows and the index sets of the equality rows, respectively. for -norm, Coefficient matrix With the new iteration point Multiply; if ,from Selecting the index set based on probability ,like ,from Selecting the index set based on probability For those already selected Indicators within Assign weights to convex combinations ; Calculate the optimal step size and optimal momentum coefficient ; Define the correction direction , For matrix transpose, For matrix of A matrix composed of rows, , For A quasi-diagonal matrix composed of diagonal elements. for The square of, For matrix The OK, for With iteration point Multiply, To obtain column vectors Subscript belongs to The elements of the vector, arranged in their original order, form a sub-column vector, which is obtained by the formula... calculate ;based on renew and The iteration terminates when the stopping condition is met; otherwise, the process from selecting and updating the index set is repeated. and ; The output module outputs the estimated solution to the linear feasibility problem.

9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the residual-driven adaptive momentum-averaged block Katzmalz solution method as described in any one of claims 1-7.

10. A computer device comprising a processor and a memory for storing a processor-executable computer program, characterized in that, When the processor executes the computer program stored in the memory, it implements the residual-driven adaptive momentum-averaged block Katzmalz solution method as described in any one of claims 1-7.

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