AMPL
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Through Solver, Open-source software and Mathematical optimization
One advantage of AMPL is the similarity of its syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization. Many modern solvers available on the NEOS Server (formerly hosted at the Argonne National Laboratory, currently hosted at the University of Wisconsin, Madison) accept AMPL input. According to the NEOS statistics AMPL is the most popular format for representing mathematical programming problems.
Features
AMPL features a mix of declarative and imperative programming styles. Formulating optimization models occurs via declarative language elements such as sets, scalar and multidimensional parameters, decision variables, objectives and constraints, which allow for concise description of most problems in the domain of mathematical optimization.
Procedures and control flow statements are available in AMPL for
- the exchange of data with external data sources such as spreadsheets, databases, XML and text files
- data pre- and post-processing tasks around optimization models
- the construction of hybrid algorithms for problem types for which no direct efficient solvers are available.
To support re-use and simplify construction of large-scale optimization problems, AMPL allows separation of model and data.
AMPL supports a wide range of problem types, among them:
- Linear programming
- Quadratic programming
- Nonlinear programming
- Mixed-integer programming
- Mixed-integer quadratic programming with or without convex quadratic constraints
- Mixed-integer nonlinear programming
- Second-order cone programming
- Global optimization
- Semidefinite programming problems with bilinear matrix inequalities
- Complementarity theory problems (MPECs) in discrete or continuous variables
- Constraint programming
AMPL invokes a solver in a separate process which has these advantages:
- User can interrupt the solution process at any time
- Solver errors do not affect the interpreter
- 32-bit version of AMPL can be used with a 64-bit solver and vice versa
Interaction with the solver is done through a well-defined nl interface.
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Availability
AMPL is available for many popular 32 & 64-bit operating systems including Linux, macOS, Solaris, AIX, and Windows. The translator is proprietary software maintained by AMPL Optimization LLC. However, several online services exist, providing free modeling and solving facilities using AMPL. A free student version with limited functionality and a free full-featured version for academic courses are also available.
AMPL can be used from within Microsoft Excel via the SolverStudio Excel add-in.
The AMPL Solver Library (ASL), which allows reading nl files and provides the automatic differentiation, is open-source. It is used in many solvers to implement AMPL connection.
Status history
This table present significant steps in AMPL history.
A sample model
A transportation problem from George Dantzig is used to provide a sample AMPL model. This problem finds the least cost shipping schedule that meets requirements at markets and supplies at factories.
<span id="mwxw"></span><span class="w" id="mwyA"> </span><span class="kd" id="mwyQ">set</span><span class="w" id="mwyg"> </span><span class="nv" id="mwyw">Plants</span><span class="p" id="mwzA">;</span> <span class="w" id="mwzQ"> </span><span class="kd" id="mwzg">set</span><span class="w" id="mwzw"> </span><span class="nv" id="mw0A">Markets</span><span class="p" id="mw0Q">;</span> <span class="w" id="mw0g"> </span><span class="c1" id="mw0w"># Capacity of plant p in cases</span> <span class="w" id="mw1A"> </span><span class="kd" id="mw1Q">param</span><span class="w" id="mw1g"> </span><span class="nv" id="mw1w">Capacity</span><span class="p" id="mw2A">{</span>p<span class="w" id="mw2Q"> </span><span class="kr" id="mw2g">in</span><span class="w" id="mw2w"> </span>Plants<span class="p" id="mw3A">};</span> <span class="w" id="mw3Q"> </span><span class="c1" id="mw3g"># Demand at market m in cases</span> <span class="w" id="mw3w"> </span><span class="kd" id="mw4A">param</span><span class="w" id="mw4Q"> </span><span class="nv" id="mw4g">Demand</span><span class="p" id="mw4w">{</span>m<span class="w" id="mw5A"> </span><span class="kr" id="mw5Q">in</span><span class="w" id="mw5g"> </span>Markets<span class="p" id="mw5w">};</span> <span class="w" id="mw6A"> </span><span class="c1" id="mw6Q"># Distance in thousands of miles</span> <span class="w" id="mw6g"> </span><span class="kd" id="mw6w">param</span><span class="w" id="mw7A"> </span><span class="nv" id="mw7Q">Distance</span><span class="p" id="mw7g">{</span>Plants<span class="p" id="mw7w">,</span><span class="w" id="mw8A"> </span>Markets<span class="p" id="mw8Q">};</span> <span class="w" id="mw8g"> </span><span class="c1" id="mw8w"># Freight in dollars per case per thousand miles</span> <span class="w" id="mw9A"> </span><span class="kd" id="mw9Q">param</span><span class="w" id="mw9g"> </span><span class="nv" id="mw9w">Freight</span><span class="p" id="mw-A">;</span> <span class="w" id="mw-Q"> </span><span class="c1" id="mw-g"># Transport cost in thousands of dollars per case</span> <span class="w" id="mw-w"> </span><span class="kd" id="mw_A">param</span><span class="w" id="mw_Q"> </span><span class="nv" id="mw_g">TransportCost</span><span class="p" id="mw_w">{</span>p<span class="w" id="mwAQA"> </span><span class="kr" id="mwAQE">in</span><span class="w" id="mwAQI"> </span>Plants<span class="p" id="mwAQM">,</span><span class="w" id="mwAQQ"> </span>m<span class="w" id="mwAQU"> </span><span class="kr" id="mwAQY">in</span><span class="w" id="mwAQc"> </span>Markets<span class="p" id="mwAQg">}</span><span class="w" id="mwAQk"> </span><span class="p" id="mwAQo">:</span><span class="o" id="mwAQs">=</span> <span class="w" id="mwAQw"> </span>Freight<span class="w" id="mwAQ0"> </span><span class="o" id="mwAQ4">*</span><span class="w" id="mwAQ8"> </span>Distance<span class="p" id="mwARA">[</span>p<span class="p" id="mwARE">,</span><span class="w" id="mwARI"> </span>m<span class="p" id="mwARM">]</span><span class="w" id="mwARQ"> </span><span class="o" id="mwARU">/</span><span class="w" id="mwARY"> </span><span class="mi" id="mwARc">1000</span><span class="p" id="mwARg">;</span> <span class="w" id="mwARk"> </span><span class="c1" id="mwARo"># Shipment quantities in cases</span> <span class="w" id="mwARs"> </span><span class="kd" id="mwARw">var</span><span class="w" id="mwAR0"> </span><span class="nv" id="mwAR4">shipment</span><span class="p" id="mwAR8">{</span>Plants<span class="p" id="mwASA">,</span><span class="w" id="mwASE"> </span>Markets<span class="p" id="mwASI">}</span><span class="w" id="mwASM"> </span><span class="o" id="mwASQ">>=</span><span class="w" id="mwASU"> </span><span class="mi" id="mwASY">0</span><span class="p" id="mwASc">;</span> <span class="w" id="mwASg"> </span><span class="c1" id="mwASk"># Total transportation costs in thousands of dollars</span> <span class="w" id="mwASo"> </span><span class="kd" id="mwASs">minimize</span><span class="w" id="mwASw"> </span><span class="nv" id="mwAS0">cost</span><span class="p" id="mwAS4">:</span> <span class="w" id="mwAS8"> </span><span class="kr" id="mwATA">sum</span><span class="p" id="mwATE">{</span>p<span class="w" id="mwATI"> </span><span class="kr" id="mwATM">in</span><span class="w" id="mwATQ"> </span>Plants<span class="p" id="mwATU">,</span><span class="w" id="mwATY"> </span>m<span class="w" id="mwATc"> </span><span class="kr" id="mwATg">in</span><span class="w" id="mwATk"> </span>Markets<span class="p" id="mwATo">}</span><span class="w" id="mwATs"> </span>TransportCost<span class="p" id="mwATw">[</span>p<span class="p" id="mwAT0">,</span><span class="w" id="mwAT4"> </span>m<span class="p" id="mwAT8">]</span><span class="w" id="mwAUA"> </span><span class="o" id="mwAUE">*</span><span class="w" id="mwAUI"> </span>shipment<span class="p" id="mwAUM">[</span>p<span class="p" id="mwAUQ">,</span><span class="w" id="mwAUU"> </span>m<span class="p" id="mwAUY">];</span> <span class="w" id="mwAUc"> </span><span class="c1" id="mwAUg"># Observe supply limit at plant p</span> <span class="w" id="mwAUk"> </span><span class="kd" id="mwAUo">s.t.</span><span class="w" id="mwAUs"> </span><span class="nv" id="mwAUw">supply</span><span class="p" id="mwAU0">{</span>p<span class="w" id="mwAU4"> </span><span class="kr" id="mwAU8">in</span><span class="w" id="mwAVA"> </span>Plants<span class="p" id="mwAVE">}:</span><span class="w" id="mwAVI"> </span><span class="kr" id="mwAVM">sum</span><span class="p" id="mwAVQ">{</span>m<span class="w" id="mwAVU"> </span><span class="kr" id="mwAVY">in</span><span class="w" id="mwAVc"> </span>Markets<span class="p" id="mwAVg">}</span><span class="w" id="mwAVk"> </span>shipment<span class="p" id="mwAVo">[</span>p<span class="p" id="mwAVs">,</span><span class="w" id="mwAVw"> </span>m<span class="p" id="mwAV0">]</span><span class="w" id="mwAV4"> </span><span class="o" id="mwAV8"><=</span><span class="w" id="mwAWA"> </span>Capacity<span class="p" id="mwAWE">[</span>p<span class="p" id="mwAWI">];</span> <span class="w" id="mwAWM"> </span><span class="c1" id="mwAWQ"># Satisfy demand at market m</span> <span class="w" id="mwAWU"> </span><span class="kd" id="mwAWY">s.t.</span><span class="w" id="mwAWc"> </span><span class="nv" id="mwAWg">demand</span><span class="p" id="mwAWk">{</span>m<span class="w" id="mwAWo"> </span><span class="kr" id="mwAWs">in</span><span class="w" id="mwAWw"> </span>Markets<span class="p" id="mwAW0">}:</span><span class="w" id="mwAW4"> </span><span class="kr" id="mwAW8">sum</span><span class="p" id="mwAXA">{</span>p<span class="w" id="mwAXE"> </span><span class="kr" id="mwAXI">in</span><span class="w" id="mwAXM"> </span>Plants<span class="p" id="mwAXQ">}</span><span class="w" id="mwAXU"> </span>shipment<span class="p" id="mwAXY">[</span>p<span class="p" id="mwAXc">,</span><span class="w" id="mwAXg"> </span>m<span class="p" id="mwAXk">]</span><span class="w" id="mwAXo"> </span><span class="o" id="mwAXs">>=</span><span class="w" id="mwAXw"> </span>Demand<span class="p" id="mwAX0">[</span>m<span class="p" id="mwAX4">];</span> <span class="w" id="mwAX8"> </span><span class="kr" id="mwAYA">data</span><span class="p" id="mwAYE">;</span> <span class="w" id="mwAYI"> </span><span class="kd" id="mwAYM">set</span><span class="w" id="mwAYQ"> </span><span class="nv" id="mwAYU">Plants</span><span class="w" id="mwAYY"> </span><span class="p" id="mwAYc">:</span><span class="o" id="mwAYg">=</span><span class="w" id="mwAYk"> </span>seattle<span class="w" id="mwAYo"> </span>san<span class="o" id="mwAYs">-</span>diego<span class="p" id="mwAYw">;</span> <span class="w" id="mwAY0"> </span><span class="kd" id="mwAY4">set</span><span class="w" id="mwAY8"> </span><span class="nv" id="mwAZA">Markets</span><span class="w" id="mwAZE"> </span><span class="p" id="mwAZI">:</span><span class="o" id="mwAZM">=</span><span class="w" id="mwAZQ"> </span>new<span class="o" id="mwAZU">-</span>york<span class="w" id="mwAZY"> </span>chicago<span class="w" id="mwAZc"> </span>topeka<span class="p" id="mwAZg">;</span> <span class="w" id="mwAZk"> </span><span class="kd" id="mwAZo">param</span><span class="w" id="mwAZs"> </span><span class="nv" id="mwAZw">Capacity</span><span class="w" id="mwAZ0"> </span><span class="p" id="mwAZ4">:</span><span class="o" id="mwAZ8">=</span> <span class="w" id="mwAaA"> </span>seattle<span class="w" id="mwAaE"> </span><span class="mi" id="mwAaI">350</span> <span class="w" id="mwAaM"> </span>san<span class="o" id="mwAaQ">-</span>diego<span class="w" id="mwAaU"> </span><span class="mi" id="mwAaY">600</span><span class="p" id="mwAac">;</span> <span class="w" id="mwAag"> </span><span class="kd" id="mwAak">param</span><span class="w" id="mwAao"> </span><span class="nv" id="mwAas">Demand</span><span class="w" id="mwAaw"> </span><span class="p" id="mwAa0">:</span><span class="o" id="mwAa4">=</span> <span class="w" id="mwAa8"> </span>new<span class="o" id="mwAbA">-</span>york<span class="w" id="mwAbE"> </span><span class="mi" id="mwAbI">325</span> <span class="w" id="mwAbM"> </span>chicago<span class="w" id="mwAbQ"> </span><span class="mi" id="mwAbU">300</span> <span class="w" id="mwAbY"> </span>topeka<span class="w" id="mwAbc"> </span><span class="mi" id="mwAbg">275</span><span class="p" id="mwAbk">;</span> <span class="w" id="mwAbo"> </span><span class="kd" id="mwAbs">param</span><span class="w" id="mwAbw"> </span><span class="nv" id="mwAb0">Distance</span><span class="w" id="mwAb4"> </span><span class="p" id="mwAb8">:</span><span class="w" id="mwAcA"> </span>new<span class="o" id="mwAcE">-</span>york<span class="w" id="mwAcI"> </span>chicago<span class="w" id="mwAcM"> </span>topeka<span class="w" id="mwAcQ"> </span><span class="p" id="mwAcU">:</span><span class="o" id="mwAcY">=</span> <span class="w" id="mwAcc"> </span>seattle<span class="w" id="mwAcg"> </span><span class="mf" id="mwAck">2.5</span><span class="w" id="mwAco"> </span><span class="mf" id="mwAcs">1.7</span><span class="w" id="mwAcw"> </span><span class="mf" id="mwAc0">1.8</span> <span class="w" id="mwAc4"> </span>san<span class="o" id="mwAc8">-</span>diego<span class="w" id="mwAdA"> </span><span class="mf" id="mwAdE">2.5</span><span class="w" id="mwAdI"> </span><span class="mf" id="mwAdM">1.8</span><span class="w" id="mwAdQ"> </span><span class="mf" id="mwAdU">1.4</span><span class="p" id="mwAdY">;</span> <span class="w" id="mwAdc"> </span><span class="kd" id="mwAdg">param</span><span class="w" id="mwAdk"> </span><span class="nv" id="mwAdo">Freight</span><span class="w" id="mwAds"> </span><span class="p" id="mwAdw">:</span><span class="o" id="mwAd0">=</span><span class="w" id="mwAd4"> </span><span class="mi" id="mwAd8">90</span><span class="p" id="mwAeA">;</span>Solvers
Here is a partial list of solvers supported by AMPL:

