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MethodAdvanced02.04
Optimization Core

Nonlinear Programming (NLP)

When objectives or constraints stop being linear.

Overview

Nonlinear Programming (NLP) covers optimization models where an objective or constraint is not purely linear. It includes smooth continuous NLP, quadratic programming, quadratically constrained quadratic programming, second-order cone programming, and nonconvex global optimization cases.

Engineering design, energy, pricing, control, finance, and machine learning all pose nonlinear models. The first practical split is convex versus nonconvex: convex NLP classes such as QP, convex QCQP, and SOCP support reliable global solutions, while nonconvex NLP requires local methods, global optimization, or careful relaxation.

Core ideas

Quadratic Programming (QP)

QP keeps the constraints linear while allowing a quadratic objective, making it a common first nonlinear extension of LP.

QCQP

QCQP allows quadratic constraints as well as a quadratic objective; convex cases can be tractable, while nonconvex cases often need relaxations or global methods.

SOCP

SOCP is a convex NLP class that represents norm bounds and many convex quadratic constraints through second-order cones.

Gradient methods

Gradient methods is a core checkpoint for Nonlinear Optimization: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Newton / quasi-Newton

Newton / quasi-Newton is a core checkpoint for Nonlinear Optimization: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

SQP

SQP is a core checkpoint for Nonlinear Optimization: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

Interior point

Solves large continuous relaxations through barrier methods and is often strong for huge sparse LPs and conic models.

Global optimization

Global optimization is a core checkpoint for Nonlinear Optimization: define it concretely, attach units or rules where possible, and test whether stakeholders interpret it the same way.

How to use it

  1. 1Write the nonlinear objective and every nonlinear constraint explicitly, including units and domains.
  2. 2Classify the model as QP, QCQP, SOCP, general convex NLP, smooth nonconvex NLP, or MINLP before selecting a solver.
  3. 3Check convexity, differentiability, scaling, bounds, and constraint qualifications on a small instance.
  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

Use this topic as a building block in nearby OR models; connect it to a concrete decision before treating it as a standalone application area.

Common pitfalls

  • Calling a model NLP without separating convex QP/QCQP/SOCP cases from nonconvex local-search cases.
  • Treating gradients, scaling, and bounds as solver details instead of modeling choices that affect the recommendation.
  • Reporting one answer without showing sensitivity to demand, capacity, costs, or behavioral assumptions.
  • Ignoring implementation details such as data quality, explainability, ownership, and how users will override bad recommendations.

Resources