assignment problem relaxation

New relaxation algorithm for three-dimensional assignment problem

Note: This article was originally prepared for the 2011 13Th IEEE Joint International Computer Science and Information Technology Conference (JICSIT)

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S-D assignment using Lagrangian relaxation

Description

[ assignments , cost , solutionGap ] = assignsd( costmatrix ) returns a table of assignments, assignments , of detections to tracks by finding a suboptimal solution to the S-D assignment problem using Lagrangian relaxation. The cost of each potential assignment is contained in the cost matrix, costmatrix . The algorithm terminates when the gap reaches below 0.01 (1 percent) or if the number of iterations reaches 100.

costmatrix is an n-dimensional cost matrix where costmatrix(i,j,k ...) defines the cost of the n-tuple (i,j,k, ...) in assignment. The index '1' on all dimensions in costmatrix represents dummy measurement or a false track and is used to complete the assignment problem. The index 1, being a dummy, can be a part of multiple n-tuples. The index can be assigned more than once. A typical cost value for costmatrix(1,1,1,1, ...) is 0.

All inputs can be single or double precision, but they all must be of the same precision.

The function also returns the solution gap, solutionGap , and the total cost of assignments, cost .

[ assignments , cost , solutionGap ] = assignsd( costmatrix , desiredGap ) also specifies the desired maximum gap, desiredGap , between the dual and the feasible solutions as a scalar. The gap controls the quality of the solution. Values usually range from 0 to 1. A value of 0 means the dual and feasible solutions are the same.

[ assignments , cost , solutionGap ] = assignsd( costmatrix , desiredGap , maxIterations ) also specifies the maximum number of iterations, maxIterations .

[ assignments , cost , solutionGap ] = assignsd( costmatrix , desiredGap , maxIterations , algorithm ) also specifies the assignment algorithm, algorithm .

collapse all

Assign Detections to Tracks Using assignsd Algorithm

Use assignsd to perform strict assignment without index 1.

Not having dummy index means that no entity is left unassigned. Therefore, define the cost matrix to be equi-dimensional.

Initialize the fullmatrix to all Inf . The fullmatix is one size larger than the cost matrix in all dimensions.

Set the inner matrix to costMatrix to force the assignments involving index 1 to have infinite cost.

Restore the actual indices.

Input Arguments

Costmatrix — cost matrix real-valued m -by- n.

Cost matrix, specified as an n-dimensional array where costmatrix(i,j,k ...) defines the cost of the n-tuple (i,j,k, ...) in an assignment. The index '1' on all dimensions in costmatrix represents a dummy measurement or a false track and is used to complete the assignment problem. The index 1, being a dummy, can be a part of multiple n-tuples. The index can be assigned more than once. A typical cost value for costmatrix(1,1,1,1, ...) is 0.

Data Types: single | double

desiredGap — Desired maximal gap 0.01 (default) | nonnegative scalar

Desired maximum gap between the dual and feasible solutions, specified as a nonnegative scalar.

Example: 0.05

maxIterations — Maximum number of iterations 100 (default) | positive integer

Maximum number of iterations, specified as a positive integer.

Example: 50

algorithm — Assignment algorithm 'auction' (default) | 'munkres' | 'jv'

Assignment algorithm for solving the 2-D assignment problem, specified as 'munkres' for the Munkres algorithm, 'jv' for the Jonker-Volgenant algorithm, or 'auction' for the Auction algorithm.

Example: 'jv'

Output Arguments

Assignments — assignment of tracks to detections p -by- n matrix.

Assignments of tracks to detections, returned as a P -by- N list of assignments. Assignments of the type [1 1 Q 1] from a four-dimensional cost matrix can be seen as a Q-1 entity from dimension 3 that was left unassigned. The cost value at (1,1,Q,1) defines the cost of not assigning the ( Q -1) th entity from dimension 3.

cost — Total cost of assignment solution positive scalar

Total cost of solutions, returned as a K -element vector where K is the number of best solutions. Each element is a scalar value summarizing the total cost of the solution to the assignment problem.

solutionGap — Solution gap positive scalar (default)

Solution gap, returned as a positive scalar. The solution gap is the duality gap achieved between the feasible and dual solution. A gap value near zero indicates the quality of solution.

The Lagrangian relaxation method computes a suboptimal solution to the S-D assignment problem. The method relaxes the S-D assignment problem to a 2-D assignment problem using a set of Lagrangian multipliers. The relaxed 2-D assignment problem is commonly known as the dual problem, which can be solved optimally using algorithms like the Munkres algorithm. Constraints are then enforced on the dual solution by solving multiple 2-D assignment problems to obtain a feasible solution to the original problem. The cost of the dual solution and the feasible solution serves as lower and upper bounds on the optimal cost, respectively. The algorithm iteratively tries to minimize the gap between the dual and feasible solutions, commonly known as the dual gap. The iteration stops when the dual gap is below a desired gap or the maximum number of iterations have reached.

When using the auction algorithm, the assignsd function uses the Heuristic Price Update algorithm to update the Lagrangian multipliers. When using the Munkres and JV algorithms, the function uses the Accelerated Subgradient Update algorithm.

For cost matrices with well-defined solutions, such as passive association with high-precision sensors, the solution gap converges to within 0.05 (5 percent) in approximately 100 iterations.

As the optimal solution is unknown, the solution gap can be non-zero even when the returned solution is optimal.

[1] Deb, S., Yeddanapudi, M., Pattipati, K., and Bar-Shalom, Y. (1997). A generalized SD assignment algorithm for multisensor-multitarget state estimation. IEEE Transactions on Aerospace and Electronic Systems, 33(2), 523-538.

[2] Blackman, Samuel, and Robert Popoli. Design and analysis of modern tracking systems . Norwood, MA: Artech House, 1999. (1999)

Extended Capabilities

C/c++ code generation generate c and c++ code using matlab® coder™..

In the generated code, assignsd returns an output with a fixed number of columns because it does not drop trailing singleton dimensions of variable-size arrays. In MATLAB, assignsd returns a variable-length output because it drops trailing singleton dimensions.

Version History

Introduced in R2018b

• assignauction | assignjv | assignkbest | assignkbestsd | assignmunkres | assignTOMHT | trackerTOMHT | trackerGNN

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Description

A general assignment problem is solved via Lagrangian Relaxation by dualizing the multiple choice constraints and solving the remaining knapsack subproblems. The data for this problem are taken from Martello. The optimal value is 223 and the optimal solution is: 1 1 4 2 3 5 1 4 3 5, where in columns 1 and 2, the variable in the first row is equal to 1, in column 3, the variable in the fourth row is equal to 1, etc...

Small Model of Types : MIP rmip

Category : GAMS Model library

Main file : gapmin.gms

Semidefinite Programming Relaxations for the Quadratic Assignment Problem

• Published: March 1998
• Volume 2 , pages 71–109, ( 1998 )

Cite this article

• Qing Zhao 1 ,
• Stefan E. Karisch 2 ,
• Franz Rendl 3 &
• Henry Wolkowicz 4  

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Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e., the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primal-dual interior-point methods.

For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting the special structure of the relaxation. See e.g., Vandenverghe and Boyd (1995) for a similar approach for solving SDP problems arising from control applications.

Numerical results are presented which indicate that the described methods yield at least competitive lower bounds.

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Semidefinite relaxation for two mixed binary quadratically constrained quadratic programs: algorithms and approximation bounds, a polynomial time constraint-reduced algorithm for semidefinite optimization problems.

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Zhao, Q., Karisch, S.E., Rendl, F. et al. Semidefinite Programming Relaxations for the Quadratic Assignment Problem. Journal of Combinatorial Optimization 2 , 71–109 (1998). https://doi.org/10.1023/A:1009795911987

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Giving Students Options for a Concept Paper in a Business Communications Course

The Communication Spotlight features innovative instructors who teach written, oral, digital/technological, kinetic, and visual communication modes.

Jennifer Hite received her BA majoring in Environmental Studies with a minor in Political Science from University of California at Santa Barbara, her MA in Communication Management from the Annenberg School of Communication at University of Southern California. She received a PhD in Organizational Behavior at UCI/The Paul Merage School of Business. Professor Hite has been an Instructor at the Annenberg School of Communication at USC, School of Business Administration at USC and UCI/The Paul Merage School of Business. She is a member of the Academy of Management, International Communication Association and the Society for Human Resources Management.

What is the assignment? 

Concept Paper: Project or Idea Pitch

Project overview: You can choose from one of two tracks for the assignment:

• Introduce a new product or
• Introduce an existing product to another country.

Track 1: Introduce a New Product

Students selecting this track will produce a concept paper and pitch that follow the requirements of the Stella Zhang New Venture Competition . By the end of the quarter, you’ll have a solid concept paper and pitch ready if you choose to compete.

Product selection, Track 1: The product must be a completely new product or a better version of an existing one that is affordable to most Americans. In addition,

• A new service or a digital product may not be used.
• If you’ve already submitted a concept paper for the New Venture Competition, you may not use the same idea or paper for MGMT 191W. However, we encourage you to use the original work you create for MGMT 191W for the competition.

Track 2: Introduce an Existing Product to Another Country

Students selecting this option will introduce an existing product to a country they are not familiar with. Here are the requirements for both the product and the country you choose.

Product selection, Track 2: The product must be an existing one that is affordable to the people in the country you’ll be introducing it to. In addition,

• It must be a consumer product ; that is, an item of common or daily use, typically bought by individuals for private consumption.
• It must be a product consumers can purchase in brick-and-mortar stores.
• Although the product you choose may already be available in the country, your goal is to find one that is not already easily available in the country .
• It cannot be a product consumers rent or that they must subscribe to, such as a meal service.
• It cannot be for commercial use only.

Country selection, Track 2 : The country you use for the report must be one you have never visited, are not from, do not have any cultural ties to, have any relatives from, or know very much about.

How does it work?

In just three pages, students must develop a complete pitch that’s designed to convince investors (Track 1) or their CEO (Track 2) to adopt their product or idea. They build a credible argument by using library resources and careful paragraph development. The paper requires them to carefully analyze the potential market characteristics as well as any competitors, and to use color to engage the reader. The skills they develop in this project are easily transportable to work assignments once they graduate.

What do students say?

“The Concept Paper was a very informative assignment. It was the combination of a research paper and a corporate pitch/report, which worked to mimic potential assignments I will have once I graduate and get a corporate job. I particularly liked that my research was catered towards a specific audience, which led to it being more refined and avoiding any unnecessary information.” – Student Response

Student Artifact: 

This paper, pitching a new product idea, engages the audience with color and in the first paragraph with an attention-getting opening. They use bullet points and numbered lists to draw the reader’s eye and to quickly summarize information. The analysis of the market potential establishes the reach of the product, backed by recent, credible research. In addition, the analysis of the product’s competitors focuses on the product’s advantages over others. The paper is concise, well-written, and well-researched.

Read the full paper here .

Why does this work?

By asking students to choose between two options for their concept paper – either introducing a new product or an old product to a new market – the assignment is essentially asking students to choose their purpose and their audience. This choice can prompt students to think about the relationship between purpose and audience and craft their writing accordingly.

Check out these resources for developing business writing assignments in your communication classes:

• Implementing Student Choice within an Assignment from University of Nebraska-Lincoln
• Business Writing Handout from UNC to help students understand typical expectations for business writing
• This particular assignment asked students to use figures in their writing. Your students might find this resource from the CEWC helpful for using tables and figures.

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3 Astros takeaways: Selloff possibilities, Yordan Alvarez’s RISP problem and 2 homegrown DFAs

As he is prone to do, Justin Verlander strapped the Houston Astros to his back on Sunday before Houston’s bats finally backed up its ace during a 9-3 win against the Detroit Tigers at Comerica Park.

The victory secured a series win and salvaged a 3-3 road trip. A critical 10-game homestand looms for a Houston team still seeking some stability. Seven of the 10 games are against American League West opponents, offering the club its best chance at gaining ground from the cellar.

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Here are three takeaways from the road trip:

Dana Brown must consider all options

Most clubhouses have at least one television always tuned to MLB Network, so general manager Dana Brown had no choice but to back a ballclub that may be listening. Saying he “can’t predict any scenario” in which the Astros sell at the trade deadline is one of the only acceptable responses to a question that isn’t going away, even after a respectable road trip.

"No. No, I can't envision that. This ballclub is too good." – Astros GM Dana Brown on if he envisions a scenario where the team is a seller #MLBNow | #Relentless pic.twitter.com/0kEEULJUoW — MLB Now (@MLBNow) May 7, 2024

Since 1901, only seven teams have started a season 15-25 and finished with a winning record. Just three of them made the playoffs, though a third wild-card spot and an underperforming American League West give the Astros more hope than other teams mired in their situation.

Brown shouldn’t declare the season dead on May 7, but refusing to acknowledge the possibility of a selloff — even privately — is almost as misguided. Before the season, both The Athletic’s Keith Law and MLB Pipeline ranked Houston’s farm system 27th out of 30 teams. Infusing more talent is crucial, and part of the reason owner Jim Crane hired Brown in the first place.

Nothing Crane has done during his ownership tenure suggests he’s about to offer the type of extension it will take to retain Kyle Tucker , who launched his league-leading 13th home run on Sunday. Alex Bregman ’s brutal start also won’t stop agent Scott Boras from seeking the sort of deal Crane has never been willing to give.

If the Astros can’t engineer a turnaround and enter July with realistic playoff chances, it would be a dereliction of duty not to dangle one or both of those players during the trade deadline. Pitchers Verlander and Ryan Pressly are obvious candidates, too, but both have no-trade clauses in their contracts and would prefer to stay in Houston.

Since his franchise’s golden era began, Crane has reiterated: “While I’m here, the window is always going to be open.” His stance means much more than anything Brown will utter across the next three months.

Barring a total collapse, it’s difficult to envision Crane softening on such a strong statement and conceding, especially while carrying the largest payroll during his ownership tenure. He resides in a city and owns a franchise well-versed in this exact situation. The last team to start 15-25 and make the playoffs? The 2005 National League champion Houston Astros.

Yordan Alvarez ’s run production problem

Since April 10, Alvarez has taken 33 plate appearances with runners in scoring position. He has one hit: a game-tying single down the right-field line during Friday’s 5-2 win against the Tigers.

Among Astros, only Yainer Diaz has taken more at-bats than Alvarez with runners in scoring position across that 28-game span. Since it began, Alvarez’s OPS has plummeted 259 points from 1.038 to .779. It never dipped below .895 last season.

The situation epitomizes the Astros’ first 40 games. Houston has the sport’s second-highest batting average but has been outscored by 13 other teams. Getting the club’s best hitter a bevy of at-bats with runners in scoring position should be its foremost goal. The Astros are accomplishing it — and Alvarez isn’t coming through.

Entering Sunday, Alvarez had a minus-5 batter run value with runners in scoring position, according to Baseball Savant. Last season, it was 28. Alvarez is hitting .157 with runners in scoring position this season. A .231 expected batting average suggests there isn’t much bad luck involved, either.

Dissecting Alvarez’s issues with runners in scoring position is difficult and also arrives with the caveat of a small sample size. His 15.2 percent whiff rate with runners in scoring position is lower than his 22.9 percent career clip while his 92.8 mph average exit velocity is in line with his 93 mph season average. A 39.4 percent hard-hit rate, however, is far lower than his usual clip.

All season, manager Joe Espada has bemoaned a lack of plate discipline during run-scoring situations. Entering Sunday’s game, Alvarez had swung at 27 of the 87 pitches he saw out of the strike zone with runners in scoring position — a 31 percent clip almost identical to his overall 30.8 percent chase rate this season. That, it should be noted, is elevated from Alvarez’s career 26.6 percent rate.

Last season, Alvarez had a 28.4 chase rate with runners in scoring position. Perhaps he, like so many in this lineup, is feeling the pressure of a poor start and trying to compensate. All 176 of Alvarez’s plate appearances have arrived from either the second or third spots in the batting order. Espada isn’t about to move him, either, putting the onus on a preseason MVP candidate to discover a way to clutch up.

Farewell to two homegrown success stories?

The Astros drafted Corey Julks and Brandon Bielak three rounds apart in 2017, part of a 42-man draft class that’s already produced 13 major-league players. Neither signed for more than $150,000 and both were overshadowed by more noticeable names taken beforehand.

Both Julks and Bielak ascended to viable major-league players: a testament to Houston’s amateur scouting and player development. The team designated both players for assignment this weekend and each stands a decent chance of catching on with another club, be it via a waiver claim or small trade.

Julks grew up in The Woodlands, a Houston suburb, before playing three seasons at the University of Houston. His inclusion on the team’s Opening Day roster last season — and subsequent April success — represented a feel-good story for both the franchise and the city that houses it.

Still, Julks is 28 and sported a career .650 OPS against major-league pitching. Joey Loperfido passed him on the organization’s outfield hierarchy, Pedro León is threatening to do the same, and top prospect Jacob Melton may be in Triple A soon.

At full strength, Houston already has two right-handed hitting outfielders on its 26-man roster: Chas McCormick and Jake Meyers . Utilityman Mauricio Dubón ’s increased time in the outfield only furthered a logjam Julks could not crack.

Julks has outperformed at Triple A and has two minor-league option years remaining, which will increase his value to 29 other teams. Teams are always searching for pitching depth and Bielak’s respectable numbers at the major-league level — a 4.65 ERA in 191 2/3 innings — may entice some clubs.

However, Bielak is out of minor-league options, one of the primary reasons he made Houston’s Opening Day roster as a swingman. The team is already bereft of pitching depth and did not want to risk losing Bielak on waivers so early in the season.

Activating Cristian Javier on Saturday forced them to confront that scenario. The Astros could have optioned Shawn Dubin to Triple A and kept Bielak on the major-league roster, but Dubin’s ability to handle multiple innings with higher-upside stuff must have appealed to an Astros team that’s already put Dubin in a few high-leverage situations this season.

Bielak never got those chances, perhaps the first sign that his days were numbered. José Urquidy ’s impending return from the injured list could force either Spencer Arrighetti or Hunter Brown to the bullpen, too, taking the place Bielak once occupied.

(Photo of Yordan Alvarez: Paul Sancya / Associated Press)

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Chandler Rome is a Staff Writer for The Athletic covering the Houston Astros. Before joining The Athletic, he covered the Astros for five years at the Houston Chronicle. He is a graduate of Louisiana State University. Follow Chandler on Twitter @ Chandler_Rome

Wildland Firefighters Tackle Recruitment Challenges

Every summer, a vast majority of the United States is shrouded in thick wildfire smoke that hangs in the air like a hazy curtain, casting an eerie filter over landscapes and transforming the once-clear skies into a somber palette of muted hues. Thousands of firefighters are dispatched to distant corners of the nation to begin wildfire suppression efforts that lasts for months. At the peak of fire season, the nation can have as many as 32,000 wildland firefighters on assignment at any given time. The United States has a wildfire problem.

As wildland fire agencies, including the U.S. Fish and Wildlife Service, strive to reduce risk of wildfire to communities, the issues of recruiting and retaining wildland firefighters pose significant challenges.

The concern over recruitment and retention grows every year. Several factors contribute to the complexity of attracting and keeping personnel within wildland fire agencies: demanding nature of the work, high risk and stress levels, burnout and mental health concerns, and competitive pay. While Congress works to address firefighter pay and mental health concerns, the U.S. Fish and Wildlife Service’s Branch of Fire Management has tackled the recruitment and retention issues by addressing the competitive job market and aging workforce in new ways.

In 2022 and 2023 a film crew hired the U.S. Fish and Wildlife Service jetted across the country to capture wildland fire footage, habitats, wildlife species, fire personnel, and wildland firefighting skills for three short recruitment videos. In November 2022, filmmakers made their first filming stop at Balcones Canyonlands National Wildlife Refuge in Texas.

Next stop for the film crew brought them to Okefenokee National Wildlife Refuge in Georgia/Florida where they were promised footage of fires along with the endangered gopher tortoise. Turns out gopher tortoises are a little camera-shy, so the crew focused on the less elusive, but also endangered, red-cockaded woodpecker before capturing footage of wildlife, amphibious fire vehicles, and the FWS helicopter on multiple prescribed fires.

From burning at the best-preserved, precipitation-based freshwater wetland ecosystem in the conterminous U.S. at Okefenokee, the film crew worked their way to Turnbull National Wildlife Refuge in Washington. The film crew captured vastly different fire behavior in the Channeled Scablands ecosystem that predominates the refuge in addition moose, deer, and sleeping porcupines.

Filming was topped off with incredible footage of airboats at Big Branch Marsh National Wildlife Refuge in Louisiana and the fire-dependent, colorful blend of rare orchids, carnivorous plants, and other unique groundcover that blankets the flat landscape of Mississippi Sandhill Crane National Wildlife Refuge in Mississippi. 

Once filming was complete, we released three recruitment films, each designed to target diverse, younger audiences to help shepherd in the next generation of wildland firefighters:

• Recruitment film #1 provides a two-minute intense peek into the excitement of working for our wildland fire program. This film is designed to grab the attention of younger generation thrill seekers looking for intensity in their lives.
• Recruitment film #2 offers an in-depth look into how the wildland fire program supports and enhances our overall mission, fire’s importance in creating sustainable and healthy ecosystems for plant and wildlife species, and the close-knit culture of our wildland fire program.
• Recruitment film #3 explores the upbeat, light, and fun aspect of working for our wildland fire program. This film is designed to appeal to those who may have an interest in wildland fire, but also a fear of what type of culture might greet them upon accepting a job in the field.

Recruitment and retention challenges require a multifaceted approach, including implementing strategies to promote a positive and inclusive workplace culture and targeting future recruits that emulate this culture. Collaborative efforts between government agencies, communities, and educational institutions are essential to building a sustainable and resilient wildland firefighting workforce. We hope these recruitment videos help connect communities, educational institutions, and the next generation of firefighters to wildland firefighting with the U.S. Fish and Wildlife Service.

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