# mathematical optimization

> study of mathematical algorithms for optimization problems

**Wikidata**: [Q141495](https://www.wikidata.org/wiki/Q141495)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Mathematical_optimization)  
**Source**: https://4ort.xyz/entity/mathematical-optimization

## Summary
Mathematical optimization is the academic discipline dedicated to the study of mathematical algorithms designed to solve optimization problems. It serves as a foundational branch of applied mathematics and computer science, encompassing subfields such as convex, discrete, and stochastic optimization to find the best solution from a set of alternatives. The field is characterized by its rigorous algorithmic approach to maximizing or minimizing objective functions under specific constraints.

## Key Facts
- **Definition**: The study of mathematical algorithms for optimization problems.
- **Classification**: It is an instance of an academic discipline and a branch of applied mathematics.
- **Subfields**: Includes convex optimization, stochastic optimization, discrete optimization, combinatorial optimization, multi-objective optimization, continuous optimization, global optimization, and robust optimization.
- **Related Concepts**: Closely linked to numerical analysis, algorithmics, slack variables, logic optimization, program optimization, and Bayesian optimization.
- **Key Algorithms & Methods**: Encompasses the Nelder–Mead method, evolution strategies, random search, and hill climbing (which is also classified as a genetic algorithm subclass).
- **Notable Prizes**: The George B. Dantzig Prize was established in 1982 (inception reference 1979) to honor contributions to the field.
- **Authority Identifiers**:
  - P227: 4043664-0
  - P244: sh85107312
  - P646: /m/0dpl_
  - P1036: 519.6, 519.7
  - P1149: QA402.5-QA402.6
  - P3417: Mathematical-Optimization
  - P8408: MathematicalOptimization
- **Aliases**: Mathematical optimisation, mathematical programming, optimization, optimisation.
- **Wikipedia Presence**: 57 sitelinks across various language editions.

## FAQs
**What are the primary sub-disciplines within mathematical optimization?**
The field branches into several specialized areas including convex optimization, discrete optimization, combinatorial optimization, and stochastic optimization. Other distinct branches include continuous optimization, global optimization, multi-objective optimization, and robust optimization, each addressing specific types of problem constraints and objective functions.

**Which specific algorithms and methods are central to this field?**
Core methodologies include the Nelder–Mead method for numerical optimization, evolution strategies, and random search techniques. The field also utilizes hill climbing, which is classified as both a search algorithm and a subclass of genetic algorithms, as well as Bayesian optimization for handling undifferentiable, black-box functions.

**Who are the key figures associated with mathematical optimization?**
Prominent contributors include George B. Dantzig, honored by the Dantzig Prize; Lev Pontryagin, a Soviet mathematician; and Philip Wolfe, known for the Frank–Wolfe algorithm. Other notable researchers include David S. Johnson, Robert E. Bixby, Hélène Frankowska, Ferenc Radó, Dimitris Bertsimas, Andrzej Piotr Ruszczyński, Arkadi Nemirovski, and Rozetta Zhilina.

**How is mathematical optimization related to computer science and engineering?**
It intersects deeply with algorithmics, logic optimization in digital electronics, and program optimization for software efficiency. The discipline also relies on numerical analysis for approximations and is a critical component of applied mathematics used in engineering, finance, and scientific modeling.

## Why It Matters
Mathematical optimization is the engine behind solving complex decision-making problems across science, engineering, and economics. By providing rigorous algorithms to maximize efficiency or minimize costs, it enables advancements in fields ranging from logistics and supply chain management to machine learning and structural design. The development of specific techniques like the Frank–Wolfe algorithm and the Nelder–Mead method has allowed researchers to tackle non-linear and high-dimensional problems that were previously intractable. Its role in logic optimization and program optimization directly impacts the performance of modern computing hardware and software, making it indispensable for technological progress. Furthermore, the field's sub-disciplines, such as multi-objective and robust optimization, allow for the handling of real-world uncertainties and conflicting goals, ensuring that solutions are not only optimal but also resilient.

## Notable For
- **Defining the Algorithmic Study**: It is the specific academic field dedicated to the study of mathematical algorithms for optimization problems.
- **Diverse Subfield Ecosystem**: It uniquely encompasses a wide range of specialized branches including convex, stochastic, discrete, combinatorial, and multi-objective optimization.
- **Integration with Genetic Algorithms**: It is the parent domain for hill climbing, which is explicitly classified as a subclass of genetic algorithms.
- **Prestigious Recognition**: It is the subject of the George B. Dantzig Prize, a major award established in 1982.
- **Cross-Disciplinary Utility**: It bridges pure mathematics with practical applications in logic optimization, program optimization, and numerical analysis.
- **Global Academic Reach**: The topic maintains a significant presence with 57 Wikipedia sitelinks and is recognized under numerous international authority identifiers (e.g., P227, P244, P1036).

## Body

### Definition and Classification
Mathematical optimization is formally defined as the study of mathematical algorithms for optimization problems. It is categorized as an academic discipline and a branch of applied mathematics. In structured knowledge graphs, it is an instance of an academic field of study and is a subclass of broader mathematical concepts. The field is also known by several aliases, including mathematical optimisation, mathematical programming, optimization, and optimisation. It is intrinsically linked to the study of algorithms and data structures under the domain of algorithmics.

### Subfields and Specializations
The discipline is divided into numerous specialized branches, each addressing different problem structures:
- **Convex Optimization**: A subfield focused on convex problems.
- **Discrete Optimization**: A branch dealing with discrete variables.
- **Combinatorial Optimization**: A subset concerned with combinatorial structures.
- **Stochastic Optimization**: A method handling uncertainty and random variables.
- **Continuous Optimization**: A branch of optimization in applied mathematics.
- **Global Optimization**: A branch of mathematics seeking global optima.
- **Multi-objective Optimization**: An area of multiple criteria decision making involving more than one objective function to be optimized simultaneously.
- **Robust Optimization**: A type of optimization designed to handle uncertainty.
- **Logic Optimization**: A process specifically used in digital electronics and integrated circuit design.
- **Program Optimization**: The process of modifying software to improve efficiency or performance.
- **Bayesian Optimization**: A technique specifically for undifferentiable, black-box functions.

### Algorithms and Methods
Various specific algorithms and techniques are central to the practice of mathematical optimization:
- **Nelder–Mead Method**: A numerical optimization algorithm.
- **Evolution Strategy**: A mathematical optimization technique.
- **Random Search**: A numerical optimization method.
- **Hill Climbing**: An optimization algorithm classified as a subclass of search algorithms and genetic algorithms. It is also known as HC, Bergsteiger-Algorithmus, Bergsteigerverfahren, and восхождение к вершине. It is distinct from physical mountain hiking and the mountain climbing problem.
- **Frank–Wolfe Algorithm**: A key method developed by Philip Wolfe for linear and nonlinear programming.
- **Speculative Execution**: An optimization technique used in computing.

### Key Figures and Contributors
The field has been shaped by numerous mathematicians and computer scientists:
- **George B. Dantzig**: Honored by the George B. Dantzig Prize (inception 1982, reference 1979).
- **Lev Pontryagin**: A Soviet mathematician (1908–1988) with citizenship in the Soviet Union and Russia.
- **Ferenc Radó**: A Hungarian mathematician (1921–1990).
- **Robert E. Bixby**: An American mathematician.
- **Hélène Frankowska**: A French mathematician with citizenship in France and Poland.
- **David S. Johnson**: An American computer scientist (1945–2016) known for contributions to algorithms, complexity theory, and mathematical optimization. He was a faculty member at Columbia University, received the Knuth Prize (2010) and Frederick W. Lanchester Prize (1979), and was an ACM and SIAM Fellow.
- **Philip Wolfe**: An American mathematician (born 1927) known for the Frank–Wolfe algorithm and recipient of the John von Neumann Theory Prize (1992).
- **Rozetta Zhilina**: A Soviet and Russian mathematician and computer scientist (1933–2003) who worked at the All-Russian Scientific Research Institute of Technical Physics and received the USSR State Prize.
- **Gleb Nosovsky**: A Russian mathematician.
- **Darinka Dentcheva**: A Bulgarian-American mathematician.
- **Dimitris Bertsimas**: A Greek American professor of management and operations research.
- **Ross Honsberger**: A Canadian mathematician (1929–2016).
- **Sergey Stechkin**: A Russian mathematician (1920–1995).
- **Andrzej Piotr Ruszczyński**: A Polish-American mathematician.
- **Adrian Lewis**: A British-Canadian mathematician.
- **Arkadi Nemirovski**: An American mathematician.

### Related Concepts and Parent Fields
Mathematical optimization is part of and related to several broader fields:
- **Applied Mathematics**: The discipline of mathematics to which optimization belongs.
- **Numerical Analysis**: The study of algorithms that use numerical approximation for problems of mathematical analysis.
- **Algorithmics**: The study of algorithms and data structures.
- **Slack Variable**: A mathematical concept used within optimization.
- **Meta-optimization**: The use of one optimization method to tune another optimization method.

### Awards and Honors
Recognition in the field is highlighted by specific awards:
- **The George B. Dantzig Prize**: An award established in 1982 (with inception references to 1979).
- **John von Neumann Theory Prize**: Received by Philip Wolfe in 1992.
- **Knuth Prize**: Received by David S. Johnson in 2010.
- **Frederick W. Lanchester Prize**: Received by David S. Johnson in 1979.
- **USSR State Prize**: Received by Rozetta Zhilina.
- **Order of the Red Banner of Labour**: Awarded to Rozetta Zhilina.
- **Medal "For Labour Valour"**: Awarded to Rozetta Zhilina.

### Institutional and Community Presence
The field is supported by a global community and various identifiers:
- **Wikipedia**: The topic has a sitelink count of 57 and is covered in multiple languages.
- **Authority Files**: It is identified by P227 (4043664-0), P244 (sh85107312), P268 (11932649z), P646 (/m/0dpl_), P691 (ph122672), P950 (XX533091), P1036 (519.6, 519.7), P1149 (QA402.5-QA402.6), P1245 (687670), P1417 (topic/optimization), P2163 (1012099), P2347 (17635), P2812 (OptimizationTheory), P3365 (ottimizzazione), P3417 (Mathematical-Optimization), P3553 (19616819), P3827 (numerical-optimization, mathematical-optimization), P3911 (15055-0), P3984 (optimization), P4223 (ottimizzazione), P5019 (optimierung-mathematik), P5844 (ottimizzazione), P6366 (126255220), P8189 (987007538691105171), P8313 (optimering), P8408 (MathematicalOptimization), P8885 (최적화 이론), P9621 (ottimizzazione), P9935 (ottimizzazione), P10283 (C2989189746, C2987595161, C126255220, C3020013979), P11514 (optimizatsiia-79076e), and P13591 (concept/daeacc1b-a6c6-41b3-9869-220ba8283304).
- **Community Tags**: The topic is active on Stack Overflow and other technical platforms.

## References

1. [Nuovo soggettario](https://thes.bncf.firenze.sbn.it/termine.php?id=21954)
2. [Source](https://www.britannica.com/science/optimization)
3. Nuovo soggettario
4. Freebase Data Dumps. 2013
5. YSO-Wikidata mapping project
6. Quora
7. National Library of Israel
8. KBpedia
9. [OpenAlex](https://docs.openalex.org/download-snapshot/snapshot-data-format)