Input Methods Overview

HPRQP supports three different ways to specify and solve quadratic programming problems. Choose the method that best fits your workflow.

Quick Comparison

Choosing Your Input Method

Use JuMP Integration if you:

• ✓ Want to build models using algebraic expressions
• ✓ Need a high-level, readable problem formulation
• ✓ Are familiar with optimization modeling languages (like AMPL, GAMS)
• ✓ Want to easily switch between different solvers
• ✓ Prefer working with variables, constraints, and objectives directly

Example use case: Building a portfolio optimization model with quadratic risk terms.

using JuMP, HPRQP
model = Model(HPRQP.Optimizer)
@variable(model, x[1:n] >= 0)
@objective(model, Min, sum(x[i]^2 for i in 1:n) + sum(c[i]*x[i] for i in 1:n))
@constraint(model, sum(x) == 1)
optimize!(model)

→ See JuMP Integration Guide

Use MPS Files if you:

• ✓ Have problems in MPS format from benchmarks (MAROS, CUTEst, etc.)
• ✓ Need to read problems generated by other optimization tools
• ✓ Want to use standard QP test sets
• ✓ Are comparing solver performance on established benchmarks
• ✓ Receive problem files from collaborators or other software

Example use case: Solving benchmark QP problems from standard test sets or problems exported from MATLAB/GAMS.

using HPRQP
model = build_from_mps("path/to/qp_problem.mps")
params = HPRQP_parameters()
result = optimize(model, params)

→ See MPS Files Guide

Use Direct API if you:

• ✓ Already have your problem in matrix form
• ✓ Need maximum performance (no modeling overhead)
• ✓ Are implementing custom algorithms or research code
• ✓ Work with problems generated from scientific computing
• ✓ Want precise control over the problem formulation and Q operator
• ✓ Are integrating HPRQP into existing numerical code

Example use case: Solving QP subproblems in a decomposition algorithm or using specialized Q operators for structured problems.

using HPRQP, SparseArrays
Q = sparse([...])  # Your quadratic term
A = sparse([...])  # Your constraint matrix
model = build_from_QAbc(Q, A, c, AL, AU, l, u)
params = HPRQP_parameters()
result = optimize(model, params)

→ See Direct API Guide

Can I Use Multiple Methods?

Yes! You can use different input methods in the same project. For example:

• Use JuMP for rapid prototyping, then switch to Direct API for production
• Validate your Direct API implementation against MPS benchmark files
• Build models in JuMP and export/import via MPS format

Understanding Q Operators

HPRQP supports different representations of the quadratic term Q:

1. Sparse Matrix - For general sparse quadratic problems
2. LASSO Operator - For L1-regularized least squares (||Ax - b||² + λ||x||₁)
3. QAP Operator - For quadratic assignment problems
4. Custom Operators - Define your own matrix-free operators

→ See Q Operators Overview

What's Next?

1. Choose your input method using the guide above
2. Understand Q operators for your problem type
3. Configure solver parameters in the Parameters guide
4. Understand the results in the Output & Results guide

Standard Form

Regardless of input method, HPRQP internally solves problems in the standard form:

\[\begin{array}{ll} \min \quad & \frac{1}{2} \langle x, Qx \rangle + \langle c, x \rangle + c_0\\ \text{s.t.} \quad & AL \leq Ax \leq AU \\ & l \leq x \leq u \end{array}\]

• JuMP automatically converts your model to this form
• MPS files are parsed into this representation
• Direct API lets you specify this form directly