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MethodAdvanced02.16
Optimization Core

Quadratically Constrained Quadratic Programming (QCQP - Beta)

Quadratic objectives with quadratic constraints.

Overview

Quadratically Constrained Quadratic Programming (QCQP) extends QP by allowing quadratic constraints as well as a quadratic objective. It is a natural NLP subtopic for models where risk, distance, energy, variance, or norm-like restrictions cannot be expressed linearly.

The key modeling distinction is convex versus nonconvex QCQP. Convex QCQP can often be solved through conic or interior-point methods, while nonconvex QCQP usually needs semidefinite relaxations, bound tightening, spatial branch-and-bound, or other global optimization machinery.

Core ideas

Quadratic constraints

Quadratic constraints limit feasible decisions using squared, bilinear, norm, variance, or energy-like expressions.

Convex QCQP

Convex QCQP has a convex objective and convex feasible set, often making it tractable through conic reformulations or interior-point methods.

Nonconvex QCQP

Nonconvex QCQP can have local optima and weak relaxations, so solver results need stronger validation and bounds.

Semidefinite relaxations

SDP relaxations lift quadratic terms into matrix variables to produce bounds or approximate solutions for hard QCQP instances.

Trust-region models

Trust-region subproblems are a classic QCQP pattern where a quadratic model is optimized inside a norm-bounded region.

How to use it

  1. 1Write each quadratic objective and constraint term explicitly, including matrix symmetry and units.
  2. 2Classify every quadratic constraint as convex, concave, or indefinite before choosing a solver.
  3. 3Look for SOCP-representable constraints such as norm bounds, then use SDP or global methods for harder nonconvex structure.
  4. 4Compare the recommendation against a baseline policy, not just against mathematical optimality.
  5. 5Document assumptions, sensitivity results, and the conditions under which the recommendation should be revisited.

Applications

Portfolio riskSignal processingRobust optimizationPower systems
  • Portfolio risk: variance and tracking-error limits are natural quadratic constraints.
  • Signal processing: beamforming and estimation models often contain quadratic power or norm limits.
  • Robust optimization: ellipsoidal uncertainty sets commonly lead to conic or quadratic constraints.
  • Power systems: AC power-flow relaxations and engineering limits can produce QCQP structure.

Common pitfalls

  • Treating nonconvex QCQP output as globally optimal without a certificate or bound.
  • Missing an SOCP reformulation that would make a convex quadratic constraint easier to solve.
  • Letting poorly scaled quadratic terms dominate numerical behavior.
  • Reporting one answer without showing sensitivity to demand, capacity, costs, or behavioral assumptions.

Resources