用戶:Alexander Misel/沙盒

一個簡單的二元線性規劃問題的圖示表示。其中包含六個不等式約束，可行解集用黃色表示，形成了一個多邊形，即二維多胞形。線性目標函數的最優解位於紅線與多邊形的交點處。這條紅線是目標函數的等值線，箭頭指示了優化的方向。

一個具有三個變量的問題的閉合可行域是一個凸多面體。目標函數的固定值所形成的表面是平面（圖中未顯示）。線性規劃問題就是要在這個多面體上找到一個點，使其位於具有最高可能值的平面上。

線性規劃（英語：Linear Programming，簡稱LP）是一種數學方法，通過線性方程或不等式描述問題的約束條件和目標，以實現最佳結果（例如利潤最大化或成本最小化）。作為最優化的一種特例，線性規劃在許多領域都有重要應用。

更嚴謹地說，線性規劃旨在優化一個線性目標函數，該函數需滿足一定的線性等式和不等式約束。其解的可行域是一個凸多胞形，這一區域由若干線性不等式描述的有限半空間的交集定義。目標函數本質上是定義在這一凸多面體上的實值仿射函數。通過線性規劃算法，可以在多胞形內找到目標函數的最大值或最小值（若解存在）。

線性規劃問題通常用標準型表達為：

${\begin{aligned}&{\text{Find a vector}}&&\mathbf {x} \\&{\text{that maximizes}}&&\mathbf {c} ^{\mathsf {T}}\mathbf {x} \\&{\text{subject to}}&&A\mathbf {x} \leq \mathbf {b} \\&{\text{and}}&&\mathbf {x} \geq \mathbf {0} .\end{aligned}}$

其中，$\mathbf {x}$是待求解的變量向量，$\mathbf {c}$和$\mathbf {b}$是已知向量，$A$是已知矩陣。需要最大化的$\mathbf {x} \mapsto \mathbf {c} ^{\mathsf {T}}\mathbf {x}$被稱為目標函數，而約束條件$A\mathbf {x} \leq \mathbf {b}$和$\mathbf {x} \geq \mathbf {0}$定義了目標函數優化範圍內的凸多面體。

線性規劃的應用覆蓋多個領域。它在數學研究中尤為常見，同時也在商業、經濟學以及某些工程問題中具有重要價值。線性規劃與特徵方程、馮·諾依曼的總體均衡模型及結構均衡模型緊密相關（詳見對偶線性規劃）。 目前，運輸、能源、電信和製造業等行業廣泛使用線性規劃模型。通過這種方法，可以高效解決規劃、路由、日程安排、任務分配和設計等各類複雜問題，為實際應用提供精確的數學支持。

History

Leonid Kantorovich

John von Neumann

線性不等式組求解問題可追溯到傅里葉的時期，他於1827年發表了一種求解方法， 這一方法後來被稱為傅里葉-莫茨金消元法。

20世紀30年代末期，蘇聯數學家康托羅維奇和美國經濟學家列昂惕夫各自獨立開展了線性規劃的應用研究。康托羅維奇致力於解決生產調度問題，列昂惕夫則專注於經濟領域的應用。然而，他們的開創性成果在相當長的時期內並未受到應有的重視。

二戰期間，線性規劃迎來了重大轉機。這一數學工具在應對戰時各種複雜挑戰時展現出獨特優勢，特別是在運輸物流、任務調度和資源分配等方面。考慮到成本和資源限制等現實約束條件，線性規劃在優化這些環節時發揮了不可替代的作用。

正是戰時的顯著成效讓線性規劃逐漸受到廣泛關注。二戰結束後，這一方法獲得了學界普遍認可，並在運籌學、經濟學等諸多領域奠定了基礎性地位。康托羅維奇和列昂惕夫在30年代末期提出的理論貢獻，最終成為線性規劃在決策優化領域廣泛應用的重要基石。

康托羅維奇的研究成果起初在蘇聯並未得到重視。 同一時期，美籍荷蘭經濟學家庫普曼斯開始用線性規劃方法處理經典經濟問題。兩位學者後來共同獲得了1975年諾貝爾經濟學獎。 1941年，希區柯克（Frank Lauren Hitchcock）將運輸問題也納入線性規劃框架，提出了一種與後來的單純形法極為相似的解法。 可惜希區柯克於1957年去世，而諾貝爾獎是不能追授的。

1946年至1947年間，丹齊格獨立開發了通用線性規劃方法，用於解決美國空軍的規劃難題。 1947年，他發明了單純形法，這是首個能夠高效解決大多數線性規劃問題的方法。 當丹齊格與馮·諾伊曼會面討論單純形法時，後者敏銳地發現這一理論與其正在研究的博弈論問題本質上是等價的，由此提出了對偶理論。 丹齊格在1948年1月5日完成的未發表報告《線性不等式定理》（A Theorem on Linear Inequalities）中對此作出了嚴格證明。 他的研究成果於1951年正式發表，此後在戰後各行業的日常規劃中得到廣泛應用。

丹齊格最初研究的是一個70人對應70個崗位的最優分配問題。若要窮舉所有可能的排列組合來尋找最佳方案，所需的計算量是天文數字，甚至超過了可觀測宇宙中的粒子總數。然而，將這一問題轉化為線性規劃模型並使用單純形法，卻能在很短時間內求得最優解。這得益於線性規劃理論大幅降低了需要檢驗的可行解數量。

1979年，哈奇揚（Leonid Khachiyan）首次證明了線性規劃問題可在多項式時間內求解。 而該領域更具突破性的理論與實踐進展出現在1984年，當時卡馬卡（Narendra Karmarkar）提出了求解線性規劃的新型內點法。

用途

線性規劃作為一個被廣泛應用的優化領域，這絕非偶然。運籌學中大量的實際問題都可以轉化為線性規劃問題。 在線性規劃領域，網絡流問題和多商品流問題等特殊案例因其重要性而催生了大量針對性的算法研究。許多其他類型的優化算法也往往通過解決線性規劃的子問題來實現其目標。從發展歷程來看，線性規劃孕育了優化理論中的諸多核心理念，包括對偶性、分解，以及凸性及其推廣的重要性等。線性規劃不僅在微觀經濟學的創立期發揮了重要作用，如今在企業管理中仍然扮演着關鍵角色，廣泛應用於規劃、生產、運輸和技術等領域。雖然現代企業面臨的管理挑戰日新月異，但在有限資源條件下實現利潤最大化和成本最小化始終是企業追求的目標。值得一提的是，谷歌也將線性規劃應用於YouTube視頻的穩定性優化。

標準型

標準型是描述線性規劃問題時最常用、最直觀的形式。其由以下三個部分組成：

• 需要最大化的線性（或仿射）目標函數

e.g. $f(x_{1},x_{2})=c_{1}x_{1}+c_{2}x_{2}$

• 問題約束條件，形式如下：

e.g.

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}&\leq b_{1}\\a_{21}x_{1}+a_{22}x_{2}&\leq b_{2}\\a_{31}x_{1}+a_{32}x_{2}&\leq b_{3}\\\end{matrix}}$

• 非負變量

e.g.

${\begin{matrix}x_{1}\geq 0\\x_{2}\geq 0\end{matrix}}$

問題通常以矩陣形式表達，形式如下：

$\max\{\,\mathbf {c} ^{\mathsf {T}}\mathbf {x} \mid \mathbf {x} \in \mathbb {R} ^{n}\land A\mathbf {x} \leq \mathbf {b} \land \mathbf {x} \geq 0\,\}$

其他形式，例如最小化問題、包含其他形式約束條件的問題以及涉及負變量的問題，均可以重寫為等價的標準型問題。

示例

農夫例子的圖形解法——在標出違反條件的區域後，無陰影區域中距離原點最遠的頂點（與虛線相交）給出了最優組合（該點位於土地和農藥線上，說明收入受到土地和農藥的限制，而不是化肥的限制）。

假設一位農民有一片面積為 L 公頃的農田，可以種植小麥或大麥，或者兩者的組合。農民擁有 F 千克的肥料和 P 千克的農藥。每公頃小麥需要 F₁ 千克肥料和 P₁ 千克農藥，而每公頃大麥需要 F₂ 千克肥料和 P₂ 千克農藥。設 S₁ 和 S₂ 分別為每公頃小麥和大麥的售價。如果用 x₁ 和 x₂ 分別表示種植小麥和大麥的面積，則通過選擇 x₁ 和 x₂ 的最佳值可以實現利潤最大化。這個問題可以表示為以下標準型的線性規劃問題：

最大化：	$S_{1}\cdot x_{1}+S_{2}\cdot x_{2}$	（最大化收益，即小麥總銷售額加大麥總銷售額，收益是「目標函數」）
Subject to:	$x_{1}+x_{2}\leq L$	（總面積限制）
	$F_{1}\cdot x_{1}+F_{2}\cdot x_{2}\leq F$	（肥料限制）
	$P_{1}\cdot x_{1}+P_{2}\cdot x_{2}\leq P$	（農藥限制）
	$x_{1}\geq 0,x_{2}\geq 0$	（種植面積不能為負）

矩陣形式表示為：

maximize ${\begin{bmatrix}S_{1}&S_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}$

subject to ${\begin{bmatrix}1&1\\F_{1}&F_{2}\\P_{1}&P_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\leq {\begin{bmatrix}L\\F\\P\end{bmatrix}},\,{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\geq {\begin{bmatrix}0\\0\end{bmatrix}}.$

增廣型（鬆弛型）

線性規劃問題可以轉換為增廣型，以便使用單純形法的通用形式求解。這種形式引入非負的鬆弛變量（slack variable），將約束中的不等式轉化為等式。此時問題可以用以下分塊矩陣形式表示：

最大化$z$：

${\begin{bmatrix}1&-\mathbf {c} ^{\mathsf {T}}&0\\0&\mathbf {A} &\mathbf {I} \end{bmatrix}}{\begin{bmatrix}z\\\mathbf {x} \\\mathbf {s} \end{bmatrix}}={\begin{bmatrix}0\\\mathbf {b} \end{bmatrix}}$

$\mathbf {x} \geq 0,\mathbf {s} \geq 0$

其中，$\mathbf {s}$是新引入的鬆弛變量，$\mathbf {x}$是決策變量，$z$是需要最大化的變量。

示例

上述例子可轉換為以下增廣型：

最大化：$S_{1}\cdot x_{1}+S_{2}\cdot x_{2}$		（目標函數）
subject to:	$x_{1}+x_{2}+x_{3}=L$	（增廣約束）
	$F_{1}\cdot x_{1}+F_{2}\cdot x_{2}+x_{4}=F$	（增廣約束）
	$P_{1}\cdot x_{1}+P_{2}\cdot x_{2}+x_{5}=P$	（增廣約束）
	$x_{1},x_{2},x_{3},x_{4},x_{5}\geq 0.$

其中$x_{3},x_{4},x_{5}$是（非負的）鬆弛變量，分別表示未使用的面積、未使用的肥料量和未使用的農藥量。

矩陣形式表示為：

最大化$z$：

$${\begin{bmatrix}1&-S_{1}&-S_{2}&0&0&0\\0&1&1&1&0&0\\0&F_{1}&F_{2}&0&1&0\\0&P_{1}&P_{2}&0&0&1\\\end{bmatrix}}{\begin{bmatrix}z\\x_{1}\\x_{2}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}}={\begin{bmatrix}0\\L\\F\\P\end{bmatrix}},\,{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}}\geq 0.$$

Duality

主條目：Dual linear program

Every linear programming problem, referred to as a primal problem, can be converted into a dual problem, which provides an upper bound to the optimal value of the primal problem. In matrix form, we can express the primal problem as:

Maximize c^Tx subject to Ax ≤ b, x ≥ 0;

with the corresponding symmetric dual problem,

Minimize b^Ty subject to A^Ty ≥ c, y ≥ 0.

An alternative primal formulation is:

Maximize c^Tx subject to Ax ≤ b;

with the corresponding asymmetric dual problem,

Minimize b^Ty subject to A^Ty = c, y ≥ 0.

There are two ideas fundamental to duality theory. One is the fact that (for the symmetric dual) the dual of a dual linear program is the original primal linear program. Additionally, every feasible solution for a linear program gives a bound on the optimal value of the objective function of its dual. The weak duality theorem states that the objective function value of the dual at any feasible solution is always greater than or equal to the objective function value of the primal at any feasible solution. The strong duality theorem states that if the primal has an optimal solution, x^*, then the dual also has an optimal solution, y^*, and c^Tx^*=b^Ty^*.

A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible. See dual linear program for details and several more examples.

Variations

Covering/packing dualities

A covering LP is a linear program of the form:

Minimize: b^Ty,

subject to: A^Ty ≥ c, y ≥ 0,

such that the matrix A and the vectors b and c are non-negative.

The dual of a covering LP is a packing LP, a linear program of the form:

Maximize: c^Tx,

subject to: Ax ≤ b, x ≥ 0,

such that the matrix A and the vectors b and c are non-negative.

Examples

Covering and packing LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms. For example, the LP relaxations of the set packing problem, the independent set problem, and the matching problem are packing LPs. The LP relaxations of the set cover problem, the vertex cover problem, and the dominating set problem are also covering LPs.

Finding a fractional coloring of a graph is another example of a covering LP. In this case, there is one constraint for each vertex of the graph and one variable for each independent set of the graph.

Complementary slackness

It is possible to obtain an optimal solution to the dual when only an optimal solution to the primal is known using the complementary slackness theorem. The theorem states:

Suppose that x = (x₁, x₂, ... , x_n) is primal feasible and that y = (y₁, y₂, ... , y_m) is dual feasible. Let (w₁, w₂, ..., w_m) denote the corresponding primal slack variables, and let (z₁, z₂, ... , z_n) denote the corresponding dual slack variables. Then x and y are optimal for their respective problems if and only if

• x_j z_j = 0, for j = 1, 2, ... , n, and
• w_i y_i = 0, for i = 1, 2, ... , m.

So if the i-th slack variable of the primal is not zero, then the i-th variable of the dual is equal to zero. Likewise, if the j-th slack variable of the dual is not zero, then the j-th variable of the primal is equal to zero.

This necessary condition for optimality conveys a fairly simple economic principle. In standard form (when maximizing), if there is slack in a constrained primal resource (i.e., there are "leftovers"), then additional quantities of that resource must have no value. Likewise, if there is slack in the dual (shadow) price non-negativity constraint requirement, i.e., the price is not zero, then there must be scarce supplies (no "leftovers").

Theory

Existence of optimal solutions

Geometrically, the linear constraints define the feasible region, which is a convex polytope. A linear function is a convex function, which implies that every local minimum is a global minimum; similarly, a linear function is a concave function, which implies that every local maximum is a global maximum.

An optimal solution need not exist, for two reasons. First, if the constraints are inconsistent, then no feasible solution exists: For instance, the constraints x ≥ 2 and x ≤ 1 cannot be satisfied jointly; in this case, we say that the LP is infeasible. Second, when the polytope is unbounded in the direction of the gradient of the objective function (where the gradient of the objective function is the vector of the coefficients of the objective function), then no optimal value is attained because it is always possible to do better than any finite value of the objective function.

Optimal vertices (and rays) of polyhedra

Otherwise, if a feasible solution exists and if the constraint set is bounded, then the optimum value is always attained on the boundary of the constraint set, by the maximum principle for convex functions (alternatively, by the minimum principle for concave functions) since linear functions are both convex and concave. However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere). For this feasibility problem with the zero-function for its objective-function, if there are two distinct solutions, then every convex combination of the solutions is a solution.

The vertices of the polytope are also called basic feasible solutions. The reason for this choice of name is as follows. Let d denote the number of variables. Then the fundamental theorem of linear inequalities implies (for feasible problems) that for every vertex x^* of the LP feasible region, there exists a set of d (or fewer) inequality constraints from the LP such that, when we treat those d constraints as equalities, the unique solution is x^*. Thereby we can study these vertices by means of looking at certain subsets of the set of all constraints (a discrete set), rather than the continuum of LP solutions. This principle underlies the simplex algorithm for solving linear programs.

Algorithms

參見：List of numerical analysis topics § Linear programming

In a linear programming problem, a series of linear constraints produces a convex feasible region of possible values for those variables. In the two-variable case this region is in the shape of a convex simple polygon.

Basis exchange algorithms

Simplex algorithm of Dantzig

The simplex algorithm, developed by George Dantzig in 1947, solves LP problems by constructing a feasible solution at a vertex of the polytope and then walking along a path on the edges of the polytope to vertices with non-decreasing values of the objective function until an optimum is reached for sure. In many practical problems, "stalling" occurs: many pivots are made with no increase in the objective function. In rare practical problems, the usual versions of the simplex algorithm may actually "cycle". To avoid cycles, researchers developed new pivoting rules.

In practice, the simplex algorithm is quite efficient and can be guaranteed to find the global optimum if certain precautions against cycling are taken. The simplex algorithm has been proved to solve "random" problems efficiently, i.e. in a cubic number of steps, which is similar to its behavior on practical problems.

However, the simplex algorithm has poor worst-case behavior: Klee and Minty constructed a family of linear programming problems for which the simplex method takes a number of steps exponential in the problem size. In fact, for some time it was not known whether the linear programming problem was solvable in polynomial time, i.e. of complexity class P.

Criss-cross algorithm

Like the simplex algorithm of Dantzig, the criss-cross algorithm is a basis-exchange algorithm that pivots between bases. However, the criss-cross algorithm need not maintain feasibility, but can pivot rather from a feasible basis to an infeasible basis. The criss-cross algorithm does not have polynomial time-complexity for linear programming. Both algorithms visit all 2^D corners of a (perturbed) cube in dimension D, the Klee–Minty cube, in the worst case.

Interior point

In contrast to the simplex algorithm, which finds an optimal solution by traversing the edges between vertices on a polyhedral set, interior-point methods move through the interior of the feasible region.

Ellipsoid algorithm, following Khachiyan

This is the first worst-case polynomial-time algorithm ever found for linear programming. To solve a problem which has n variables and can be encoded in L input bits, this algorithm runs in $O(n^{6}L)$ time. Leonid Khachiyan solved this long-standing complexity issue in 1979 with the introduction of the ellipsoid method. The convergence analysis has (real-number) predecessors, notably the iterative methods developed by Naum Z. Shor and the approximation algorithms by Arkadi Nemirovski and D. Yudin.

Projective algorithm of Karmarkar

主條目：Karmarkar's algorithm

Khachiyan's algorithm was of landmark importance for establishing the polynomial-time solvability of linear programs. The algorithm was not a computational break-through, as the simplex method is more efficient for all but specially constructed families of linear programs.

However, Khachiyan's algorithm inspired new lines of research in linear programming. In 1984, N. Karmarkar proposed a projective method for linear programming. Karmarkar's algorithm improved on Khachiyan's worst-case polynomial bound (giving $O(n^{3.5}L)$). Karmarkar claimed that his algorithm was much faster in practical LP than the simplex method, a claim that created great interest in interior-point methods. Since Karmarkar's discovery, many interior-point methods have been proposed and analyzed.

Vaidya's 87 algorithm

In 1987, Vaidya proposed an algorithm that runs in $O(n^{3})$ time.

Vaidya's 89 algorithm

In 1989, Vaidya developed an algorithm that runs in $O(n^{2.5})$ time. Formally speaking, the algorithm takes $O((n+d)^{1.5}nL)$ arithmetic operations in the worst case, where $d$ is the number of constraints, $n$ is the number of variables, and $L$ is the number of bits.

Input sparsity time algorithms

In 2015, Lee and Sidford showed that linear programming can be solved in ${\tilde {O}}((nnz(A)+d^{2}){\sqrt {d}}L)$ time, where ${\tilde {O}}$ denotes the soft O notation, and $nnz(A)$ represents the number of non-zero elements, and it remains taking $O(n^{2.5}L)$ in the worst case.

Current matrix multiplication time algorithm

In 2019, Cohen, Lee and Song improved the running time to ${\tilde {O}}((n^{\omega }+n^{2.5-\alpha /2}+n^{2+1/6})L)$ time, $\omega$ is the exponent of matrix multiplication and $\alpha$ is the dual exponent of matrix multiplication. $\alpha$ is (roughly) defined to be the largest number such that one can multiply an $n\times n$ matrix by a $n\times n^{\alpha }$ matrix in $O(n^{2})$ time. In a followup work by Lee, Song and Zhang, they reproduce the same result via a different method. These two algorithms remain ${\tilde {O}}(n^{2+1/6}L)$ when $\omega =2$ and $\alpha =1$. The result due to Jiang, Song, Weinstein and Zhang improved ${\tilde {O}}(n^{2+1/6}L)$ to ${\tilde {O}}(n^{2+1/18}L)$.

Comparison of interior-point methods and simplex algorithms

The current opinion is that the efficiencies of good implementations of simplex-based methods and interior point methods are similar for routine applications of linear programming. However, for specific types of LP problems, it may be that one type of solver is better than another (sometimes much better), and that the structure of the solutions generated by interior point methods versus simplex-based methods are significantly different with the support set of active variables being typically smaller for the latter one.

Open problems and recent work

There are several open problems in the theory of linear programming, the solution of which would represent fundamental breakthroughs in mathematics and potentially major advances in our ability to solve large-scale linear programs.

• Does LP admit a strongly polynomial-time algorithm?
• Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution?
• Does LP admit a polynomial-time algorithm in the real number (unit cost) model of computation?

This closely related set of problems has been cited by Stephen Smale as among the 18 greatest unsolved problems of the 21st century. In Smale's words, the third version of the problem "is the main unsolved problem of linear programming theory." While algorithms exist to solve linear programming in weakly polynomial time, such as the ellipsoid methods and interior-point techniques, no algorithms have yet been found that allow strongly polynomial-time performance in the number of constraints and the number of variables. The development of such algorithms would be of great theoretical interest, and perhaps allow practical gains in solving large LPs as well.

Although the Hirsch conjecture was recently disproved for higher dimensions, it still leaves the following questions open.

• Are there pivot rules which lead to polynomial-time simplex variants?
• Do all polytopal graphs have polynomially bounded diameter?

These questions relate to the performance analysis and development of simplex-like methods. The immense efficiency of the simplex algorithm in practice despite its exponential-time theoretical performance hints that there may be variations of simplex that run in polynomial or even strongly polynomial time. It would be of great practical and theoretical significance to know whether any such variants exist, particularly as an approach to deciding if LP can be solved in strongly polynomial time.

The simplex algorithm and its variants fall in the family of edge-following algorithms, so named because they solve linear programming problems by moving from vertex to vertex along edges of a polytope. This means that their theoretical performance is limited by the maximum number of edges between any two vertices on the LP polytope. As a result, we are interested in knowing the maximum graph-theoretical diameter of polytopal graphs. It has been proved that all polytopes have subexponential diameter. The recent disproof of the Hirsch conjecture is the first step to prove whether any polytope has superpolynomial diameter. If any such polytopes exist, then no edge-following variant can run in polynomial time. Questions about polytope diameter are of independent mathematical interest.

Simplex pivot methods preserve primal (or dual) feasibility. On the other hand, criss-cross pivot methods do not preserve (primal or dual) feasibility – they may visit primal feasible, dual feasible or primal-and-dual infeasible bases in any order. Pivot methods of this type have been studied since the 1970s. Essentially, these methods attempt to find the shortest pivot path on the arrangement polytope under the linear programming problem. In contrast to polytopal graphs, graphs of arrangement polytopes are known to have small diameter, allowing the possibility of strongly polynomial-time criss-cross pivot algorithm without resolving questions about the diameter of general polytopes.

Integer unknowns

If all of the unknown variables are required to be integers, then the problem is called an integer programming (IP) or integer linear programming (ILP) problem. In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in many practical situations (those with bounded variables) NP-hard. 0–1 integer programming or binary integer programming (BIP) is the special case of integer programming where variables are required to be 0 or 1 (rather than arbitrary integers). This problem is also classified as NP-hard, and in fact the decision version was one of Karp's 21 NP-complete problems.

If only some of the unknown variables are required to be integers, then the problem is called a mixed integer (linear) programming (MIP or MILP) problem. These are generally also NP-hard because they are even more general than ILP programs.

There are however some important subclasses of IP and MIP problems that are efficiently solvable, most notably problems where the constraint matrix is totally unimodular and the right-hand sides of the constraints are integers or – more general – where the system has the total dual integrality (TDI) property.

Advanced algorithms for solving integer linear programs include:

• cutting-plane method
• Branch and bound
• Branch and cut
• Branch and price
• if the problem has some extra structure, it may be possible to apply delayed column generation.

Such integer-programming algorithms are discussed by Padberg and in Beasley.

Integral linear programs

更多資訊：Integral polytope

A linear program in real variables is said to be integral if it has at least one optimal solution which is integral, i.e., made of only integer values. Likewise, a polyhedron $P=\{x\mid Ax\geq 0\}$ is said to be integral if for all bounded feasible objective functions c, the linear program $\{\max cx\mid x\in P\}$ has an optimum $x^{*}$ with integer coordinates. As observed by Edmonds and Giles in 1977, one can equivalently say that the polyhedron $P$ is integral if for every bounded feasible integral objective function c, the optimal value of the linear program $\{\max cx\mid x\in P\}$ is an integer.

Integral linear programs are of central importance in the polyhedral aspect of combinatorial optimization since they provide an alternate characterization of a problem. Specifically, for any problem, the convex hull of the solutions is an integral polyhedron; if this polyhedron has a nice/compact description, then we can efficiently find the optimal feasible solution under any linear objective. Conversely, if we can prove that a linear programming relaxation is integral, then it is the desired description of the convex hull of feasible (integral) solutions.

Terminology is not consistent throughout the literature, so one should be careful to distinguish the following two concepts,

• in an integer linear program, described in the previous section, variables are forcibly constrained to be integers, and this problem is NP-hard in general,
• in an integral linear program, described in this section, variables are not constrained to be integers but rather one has proven somehow that the continuous problem always has an integral optimal value (assuming c is integral), and this optimal value may be found efficiently since all polynomial-size linear programs can be solved in polynomial time.

One common way of proving that a polyhedron is integral is to show that it is totally unimodular. There are other general methods including the integer decomposition property and total dual integrality. Other specific well-known integral LPs include the matching polytope, lattice polyhedra, submodular flow polyhedra, and the intersection of two generalized polymatroids/g-polymatroids – e.g. see Schrijver 2003.

Solvers and scripting (programming) languages

Permissive licenses:

Name	License	Brief info
Gekko	MIT License	Open-source library for solving large-scale LP, QP, QCQP, NLP, and MIP optimization
GLOP	Apache v2	Google's open-source linear programming solver
JuMP	MPL License	Open-source modeling language with solvers for large-scale LP, QP, QCQP, SDP, SOCP, NLP, and MIP optimization
Pyomo	BSD	An open-source modeling language for large-scale linear, mixed integer and nonlinear optimization
SCIP	Apache v2	A general-purpose constraint integer programming solver with an emphasis on MIP. Compatible with Zimpl modelling language.
SuanShu	Apache v2	An open-source suite of optimization algorithms to solve LP, QP, SOCP, SDP, SQP in Java

Copyleft (reciprocal) licenses:

Name	License	Brief info
ALGLIB	GPL 2+	An LP solver from ALGLIB project (C++, C#, Python)
Cassowary constraint solver	LGPL	An incremental constraint solving toolkit that efficiently solves systems of linear equalities and inequalities
CLP	CPL	An LP solver from COIN-OR
glpk	GPL	GNU Linear Programming Kit, an LP/MILP solver with a native C API and numerous (15) third-party wrappers for other languages. Specialist support for flow networks. Bundles the AMPL-like GNU MathProg modelling language and translator.
lp solve	LGPL v2.1	An LP and MIP solver featuring support for the MPS format and its own "lp" format, as well as custom formats through its "eXternal Language Interface" (XLI). Translating between model formats is also possible.
Qoca	GPL	A library for incrementally solving systems of linear equations with various goal functions
R-Project	GPL	A programming language and software environment for statistical computing and graphics

MINTO (Mixed Integer Optimizer, an integer programming solver which uses branch and bound algorithm) has publicly available source code but is not open source.

Proprietary licenses:

Name	Brief info
AIMMS	A modeling language that allows to model linear, mixed integer, and nonlinear optimization models. It also offers a tool for constraint programming. Algorithm, in the forms of heuristics or exact methods, such as Branch-and-Cut or Column Generation, can also be implemented. The tool calls an appropriate solver such as CPLEX or similar, to solve the optimization problem at hand. Academic licenses are free of charge.
ALGLIB	A commercial edition of the copyleft licensed library. C++, C#, Python.
AMPL	A popular modeling language for large-scale linear, mixed integer and nonlinear optimisation with a free student limited version available (500 variables and 500 constraints).
Analytica	A general modeling language and interactive development environment. Its influence diagrams enable users to formulate problems as graphs with nodes for decision variables, objectives, and constraints. Analytica Optimizer Edition includes linear, mixed integer, and nonlinear solvers and selects the solver to match the problem. It also accepts other engines as plug-ins, including XPRESS, Gurobi, Artelys Knitro, and MOSEK.
APMonitor	API to MATLAB and Python. Solve example Linear Programming (LP) problems through MATLAB, Python, or a web-interface.
CPLEX	Popular solver with an API for several programming languages, and also has a modelling language and works with AIMMS, AMPL, GAMS, MPL, OpenOpt, OPL Development Studio, and TOMLAB. Free for academic use.
Excel Solver Function	A nonlinear solver adjusted to spreadsheets in which function evaluations are based on the recalculating cells. Basic version available as a standard add-on for Excel.
FortMP
GAMS
Gurobi Optimizer
IMSL Numerical Libraries	Collections of math and statistical algorithms available in C/C++, Fortran, Java and C#/.NET. Optimization routines in the IMSL Libraries include unconstrained, linearly and nonlinearly constrained minimizations, and linear programming algorithms.
LINDO	Solver with an API for large scale optimization of linear, integer, quadratic, conic and general nonlinear programs with stochastic programming extensions. It offers a global optimization procedure for finding guaranteed globally optimal solution to general nonlinear programs with continuous and discrete variables. It also has a statistical sampling API to integrate Monte-Carlo simulations into an optimization framework. It has an algebraic modeling language (LINGO) and allows modeling within a spreadsheet (What'sBest).
Maple	A general-purpose programming-language for symbolic and numerical computing.
MATLAB	A general-purpose and matrix-oriented programming-language for numerical computing. Linear programming in MATLAB requires the Optimization Toolbox in addition to the base MATLAB product; available routines include INTLINPROG and LINPROG
Mathcad	A WYSIWYG math editor. It has functions for solving both linear and nonlinear optimization problems.
Mathematica	A general-purpose programming-language for mathematics, including symbolic and numerical capabilities.
MOSEK	A solver for large scale optimization with API for several languages (C++, java, .net, Matlab and python).
NAG Numerical Library	A collection of mathematical and statistical routines developed by the Numerical Algorithms Group for multiple programming languages (C, C++, Fortran, Visual Basic, Java and C#) and packages (MATLAB, Excel, R, LabVIEW). The Optimization chapter of the NAG Library includes routines for linear programming problems with both sparse and non-sparse linear constraint matrices, together with routines for the optimization of quadratic, nonlinear, sums of squares of linear or nonlinear functions with nonlinear, bounded or no constraints. The NAG Library has routines for both local and global optimization, and for continuous or integer problems.
OptimJ	A Java-based modeling language for optimization with a free version available.
SAS/OR	A suite of solvers for Linear, Integer, Nonlinear, Derivative-Free, Network, Combinatorial and Constraint Optimization; the Algebraic modeling language OPTMODEL; and a variety of vertical solutions aimed at specific problems/markets, all of which are fully integrated with the SAS System.
XPRESS	Solver for large-scale linear programs, quadratic programs, general nonlinear and mixed-integer programs. Has API for several programming languages, also has a modelling language Mosel and works with AMPL, GAMS. Free for academic use.
VisSim	A visual block diagram language for simulation of dynamical systems.

See also

• Convex programming
• Dynamic programming
• Expected shortfall § Optimization of expected shortfall
• Input–output model
• Job shop scheduling
• Least absolute deviations
• Least-squares spectral analysis
• Linear algebra
• Linear production game
• Linear-fractional programming (LFP)
• LP-type problem
• Mathematical programming
• Nonlinear programming
• Odds algorithm used to solve optimal stopping problems
• Oriented matroid
• Quadratic programming, a superset of linear programming
• Semidefinite programming
• Shadow price
• Simplex algorithm, used to solve LP problems

References

• Kantorovich, L. V. Об одном эффективном методе решения некоторых классов экстремальных проблем [A new method of solving some classes of extremal problems]. Doklady Akad Sci SSSR. 1940, 28: 211–214. 
• F. L. Hitchcock: The distribution of a product from several sources to numerous localities, Journal of Mathematics and Physics, 20, 1941, 224–230.
• G.B Dantzig: Maximization of a linear function of variables subject to linear inequalities, 1947. Published pp. 339–347 in T.C. Koopmans (ed.):Activity Analysis of Production and Allocation, New York-London 1951 (Wiley & Chapman-Hall)
• J. E. Beasley, editor. Advances in Linear and Integer Programming. Oxford Science, 1996. (Collection of surveys)
• Bland, Robert G. New Finite Pivoting Rules for the Simplex Method. Mathematics of Operations Research. 1977, 2 (2): 103–107. JSTOR 3689647. doi:10.1287/moor.2.2.103. 
• Borgwardt, Karl-Heinz. The Simplex Algorithm: A Probabilistic Analysis. Algorithms and Combinatorics 1. Springer-Verlag. 1987.  (Average behavior on random problems)
• Richard W. Cottle, ed. The Basic George B. Dantzig. Stanford Business Books, Stanford University Press, Stanford, California, 2003. (Selected papers by George B. Dantzig)
• George B. Dantzig and Mukund N. Thapa. 1997. Linear programming 1: Introduction. Springer-Verlag.
• Dantzig, George B.; Thapa, Mukund N. Linear Programming 2: Theory and Extensions. Springer-Verlag. 2003.  (Comprehensive, covering e.g. pivoting and interior-point algorithms, large-scale problems, decomposition following Dantzig–Wolfe and Benders, and introducing stochastic programming.)
• Edmonds, Jack; Giles, Rick. A Min-Max Relation for Submodular Functions on Graphs. Studies in Integer Programming. Annals of Discrete Mathematics 1. 1977: 185–204. ISBN 978-0-7204-0765-5. doi:10.1016/S0167-5060(08)70734-9. 
• Fukuda, Komei; Terlaky, Tamás. Thomas M. Liebling; Dominique de Werra , 編. Criss-cross methods: A fresh view on pivot algorithms. Mathematical Programming, Series B. 1997, 79 (1–3): 369–395. CiteSeerX 10.1.1.36.9373 . MR 1464775. S2CID 2794181. doi:10.1007/BF02614325. 
• Gondzio, Jacek; Terlaky, Tamás. 3 A computational view of interior point methods. J. E. Beasley (編). Advances in linear and integer programming. Oxford Lecture Series in Mathematics and its Applications 4. New York: Oxford University Press. 1996: 103–144. MR 1438311. Postscript file at website of Gondzio and at McMaster University website of Terlaky. 
• Murty, Katta G. Linear programming. New York: John Wiley & Sons, Inc. 1983: xix+482. ISBN 978-0-471-09725-9. MR 0720547. (comprehensive reference to classical approaches). 
• Evar D. Nering and Albert W. Tucker, 1993, Linear Programs and Related Problems, Academic Press. (elementary)
• Padberg, M. Linear Optimization and Extensions, Second Edition. Springer-Verlag. 1999.  (carefully written account of primal and dual simplex algorithms and projective algorithms, with an introduction to integer linear programming – featuring the traveling salesman problem for Odysseus.)
• Papadimitriou, Christos H.; Steiglitz, Kenneth. Combinatorial Optimization: Algorithms and Complexity Corrected republication with a new preface. Dover.  (computer science)
• Todd, Michael J. The many facets of linear programming. Mathematical Programming. February 2002, 91 (3): 417–436. S2CID 6464735. doi:10.1007/s101070100261.  (Invited survey, from the International Symposium on Mathematical Programming.)
• Vanderbei, Robert J. Linear Programming: Foundations and Extensions. Springer Verlag. 2001. 
• Vazirani, Vijay V. Approximation Algorithms. Springer-Verlag. 2001. ISBN 978-3-540-65367-7.  (Computer science)

Further reading

• Dmitris Alevras and Manfred W. Padberg, Linear Optimization and Extensions: Problems and Solutions, Universitext, Springer-Verlag, 2001. (Problems from Padberg with solutions.)
• de Berg, Mark; van Kreveld, Marc; Overmars, Mark; Schwarzkopf, Otfried. Computational Geometry 2nd revised. Springer-Verlag. 2000. ISBN 978-3-540-65620-3.  含有內容需登入查看的頁面 (link) Chapter 4: Linear Programming: pp. 63–94. Describes a randomized half-plane intersection algorithm for linear programming.
• Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. 1979. ISBN 978-0-7167-1045-5.  A6: MP1: INTEGER PROGRAMMING, pg.245. (computer science, complexity theory)
• Template:Cite Gartner Matousek 2006 (elementary introduction for mathematicians and computer scientists)
• Cornelis Roos, Tamás Terlaky, Jean-Philippe Vial, Interior Point Methods for Linear Optimization, Second Edition, Springer-Verlag, 2006. (Graduate level)
• Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency. Springer. 2003. 
• Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0-471-98232-6 (mathematical)
• Gerard Sierksma; Yori Zwols. Linear and Integer Optimization: Theory and Practice. CRC Press. 2015. ISBN 978-1-498-71016-9. 
• Gerard Sierksma; Diptesh Ghosh. Networks in Action; Text and Computer Exercises in Network Optimization. Springer. 2010. ISBN 978-1-4419-5512-8.  (linear optimization modeling)
• H. P. Williams, Model Building in Mathematical Programming, Fifth Edition, 2013. (Modeling)
• Stephen J. Wright, 1997, Primal-Dual Interior-Point Methods, SIAM. (Graduate level)
• Yinyu Ye, 1997, Interior Point Algorithms: Theory and Analysis, Wiley. (Advanced graduate-level)
• Ziegler, Günter M., Chapters 1–3 and 6–7 in Lectures on Polytopes, Springer-Verlag, New York, 1994. (Geometry)

External links

• Guidance On Formulating LP Problems
• Mathematical Programming Glossary
• The Linear Programming FAQ
• Benchmarks For Optimisation Software