Optimization Functions

Background

Optimization is the mathematical discipline concerned with finding the “best” solution from a set of feasible alternatives. At its core, an optimization problem consists of three components: an objective function to minimize or maximize, a set of decision variables that can be adjusted, and a collection of constraints that define the feasible region. This framework is remarkably versatile and appears throughout science, engineering, economics, and business operations.

In operations research, optimization drives logistics, supply chain management, and resource allocation—determining the most efficient routes for delivery trucks, optimal inventory levels, or how to schedule staff across shifts. In economics and finance, portfolio optimization balances risk and return, while pricing models find equilibrium points that maximize profit subject to demand curves and competitive pressures. Engineering applications range from structural design (minimizing material usage while meeting strength requirements) to control systems (tuning parameters to minimize error or energy consumption). In data science and machine learning, optimization underpins model training: finding weights and parameters that minimize prediction error on training datasets.

Optimization problems are classified by their mathematical structure:

• Linear programming (LP) problems have linear objectives and linear constraints; they are well-understood and solved efficiently by simplex or interior-point methods.
• Integer and mixed-integer programming (MIP/MILP) extends LP by requiring some or all variables to take discrete (integer) values, enabling modeling of yes/no decisions and discrete resource allocations.
• Nonlinear programming (NLP) handles smooth but nonlinear objectives and constraints; local optimization methods (gradient descent, Newton’s method, trust-region algorithms) are effective when good starting points are available.
• Local optimization focuses on finding a local minimum (or maximum) of an objective using gradient-based methods (e.g., BFGS, L-BFGS, Newton, trust-region) or derivative-free methods (e.g., Nelder–Mead, Powell). It’s efficient for smooth or bound-constrained problems when a reasonable initial guess is available and is often used as a subroutine inside global optimization workflows.
• Global optimization addresses problems with multiple local optima, using stochastic search (genetic algorithms, simulated annealing, differential evolution) or deterministic methods (branch-and-bound, interval methods) to explore the entire search space.

Excel provides several built-in optimization capabilities, most notably the Solver Add-in, which offers algorithms for LP, NLP, and integer programming on moderate-sized models. Solver allows users to designate a target cell (objective), changing cells (decision variables), and constraint cells, then invokes appropriate algorithms to find solutions. For simple one-variable problems, Goal Seek provides a basic root-finding interface. The Analysis ToolPak supports regression and ANOVA, which are fundamentally least-squares problems. However, Excel’s built-in tools have limitations: model size constraints (200 decision variables in the standard Solver), limited algorithm choices, and lack of advanced features like stochastic programming or robust optimization. For large-scale, complex, or specialized problems, users often turn to third-party add-ins or standalone optimization software.

Assignment Problems

Assignment problems are a fundamental class of combinatorial optimization where n tasks must be assigned to n agents (or resources) in a one-to-one mapping that minimizes total cost or maximizes total benefit. These problems arise frequently in operations research and management science: assigning workers to jobs, machines to tasks, facilities to locations, or resources to projects. The mathematical structure is typically represented as a cost matrix C where C[i,j] represents the cost of assigning agent i to task j.

Linear assignment problems (LAP) assume that the total cost is simply the sum of individual assignment costs, with no interaction terms. This is the classic assignment problem solvable in polynomial time (O(n³)) using the Hungarian algorithm or successive shortest path methods. Common applications include task scheduling, workforce optimization, and matching problems in economics (e.g., stable marriage problems, kidney exchange).

Quadratic assignment problems (QAP) introduce interaction terms: the cost of assigning facility i to location a and facility j to location b depends on both the distance between locations a and b and the flow between facilities i and j. QAP is NP-hard and arises in facility layout optimization (minimizing material handling costs in a factory floor layout), circuit board design, and keyboard layout optimization. The implementations here focus on numeric cost matrices and both linear and quadratic formulations, leveraging SciPy’s efficient algorithms.

Typical use cases:

• Workforce scheduling: Assign employees to shifts or projects minimizing labor costs while respecting skills and preferences
• Resource allocation: Distribute computing jobs across servers to balance load and minimize latency
• Facility location: Place warehouses or service centers to minimize transportation and operation costs
• Production planning: Assign manufacturing orders to production lines optimizing throughput and setup times

The following Python functions are available in Excel:

Global Optimization

Global optimization addresses the challenge of finding the absolute minimum (or maximum) of a function over its entire domain, particularly when the objective function has multiple local optima. Unlike local optimization methods that converge to the nearest local minimum from a starting point, global optimization algorithms must explore the entire feasible region to avoid being trapped in suboptimal solutions. This is fundamentally more difficult—many global optimization problems are NP-hard—but the payoff is finding provably optimal or near-optimal solutions.

Stochastic global search methods use randomness to escape local minima and explore diverse regions of the search space:

• Differential Evolution is a population-based evolutionary algorithm that maintains a population of candidate solutions and evolves them through mutation, crossover, and selection operations. It is particularly effective for continuous, multimodal problems and requires minimal parameter tuning. Common applications include parameter estimation in complex models, calibrating simulation models, and engineering design optimization.
• Dual Annealing combines classical simulated annealing (which accepts uphill moves with decreasing probability to escape local minima) with local search heuristics and generalized simulated annealing schedules. It is robust across a wide range of problem types and balances exploration and exploitation effectively.
• Basin-hopping iteratively applies local minimization followed by random perturbations, accepting new positions based on the Metropolis criterion. It is especially suited for problems with many local minima separated by barriers, such as molecular conformation and protein folding.

Deterministic global methods provide mathematical guarantees or systematic coverage:

• Simplicial Homology Global Optimization (SHGO) uses topological methods and Lipschitz bounds to systematically search the feasible region, partitioning it into simplices and pruning regions that provably cannot contain the global optimum. It is particularly effective when the objective function is relatively smooth and Lipschitz continuous.
• Brute-force grid search evaluates the objective on a dense grid of points and returns the best, guaranteeing that no region smaller than the grid spacing is missed. While computationally expensive, it is conceptually simple, highly parallelizable, and useful for problems with few variables or when a coarse global picture is needed before local refinement.

These global methods are suitable for multimodal objectives arising in:

• Engineering design: finding optimal geometries, material compositions, or process parameters where multiple design concepts may exist
• Calibration and inverse problems: fitting complex simulation models to observed data when parameter spaces have multiple plausible regions
• Chemical and materials science: molecular conformation, crystal structure prediction, and reaction pathway optimization
• Machine learning hyperparameter tuning: exploring hyperparameter spaces with non-convex validation error surfaces

The following Python functions are available in Excel:

Linear Programming

Linear programming (LP) is a mature and powerful optimization framework for problems where both the objective function and all constraints are linear in the decision variables. The standard form is: minimize cᵀx subject to Ax ≤ b, Aeq·x = beq, and bounds l ≤ x ≤ u. LP has remarkable theoretical properties: the feasible region is a convex polyhedron, and if an optimal solution exists it occurs at a vertex (extreme point) of this polyhedron. This geometric structure enables efficient algorithms—the simplex method and interior-point methods—that reliably solve LP problems with thousands or even millions of variables and constraints.

Applications of linear programming span nearly every industry and domain:

• Supply chain and logistics: transportation problems (minimizing shipping costs), production planning (maximizing output given resource constraints), inventory optimization, and blending problems (e.g., diet optimization, petroleum refining)
• Finance: portfolio optimization with linear constraints, asset-liability management, arbitrage detection
• Operations research: workforce scheduling, cutting stock problems (minimizing waste when cutting raw materials), network flow optimization (maximum flow, minimum cost flow)
• Resource allocation: capital budgeting, allocating limited resources (budget, personnel, equipment) across competing projects or activities

The LINEAR_PROG function provides access to SciPy’s linprog, which implements the revised simplex method (default), interior-point methods (for large-scale problems), and the simplex method. Users specify the objective coefficients, constraint matrices, and bounds, and the solver returns optimal decision variable values and the optimal objective value along with solution status and sensitivity information (dual values, slack variables).

Mixed-integer linear programming (MILP) extends LP by allowing some or all decision variables to take only integer (or binary) values. This seemingly small change dramatically increases modeling power and computational difficulty. Integer variables enable modeling discrete decisions: yes/no choices (build a facility or not), selection from a set of alternatives (choose one machine from several options), sequencing and scheduling (task ordering), and logical constraints. MILP is NP-hard in general, and solving large MILP models requires sophisticated branch-and-bound, branch-and-cut, or branch-and-price algorithms. SciPy’s milp function provides a basic MILP solver using the HiGHS engine, suitable for moderate-sized problems (hundreds to thousands of integer variables). For large-scale industrial MILP problems, commercial solvers (Gurobi, CPLEX, Xpress) offer significantly better performance through advanced presolve, cutting planes, and heuristics.

Common MILP applications:

• Facility location: selecting sites for warehouses, plants, or service centers from a discrete set of candidate locations
• Project selection and capital budgeting: choosing which projects to fund subject to budget constraints (0-1 knapsack problems)
• Production scheduling: determining batch sizes, timing, and sequencing of production runs
• Network design: selecting edges in a network to minimize cost while ensuring connectivity or capacity requirements (e.g., telecommunications network design, power grid expansion)
• Combinatorial optimization: traveling salesman problem, vehicle routing, bin packing, set covering and partitioning

The following Python functions are available in Excel:

Local Optimization

Local optimization methods seek to find a local minimum (or maximum) of an objective function—a point where the function value is lower than at all nearby points. These methods are the workhorses of continuous optimization and are highly effective when the objective is smooth (differentiable) and a reasonable starting point is available. Local optimization does not guarantee finding the global optimum, but for well-behaved problems (convex objectives, or nonconvex objectives with good initialization), local methods often find satisfactory solutions quickly and reliably.

Gradient-based methods leverage derivative information (first and optionally second derivatives) to guide the search direction. The most common strategies include:

• Gradient descent and conjugate gradient: use the gradient (first derivative) to move in the direction of steepest descent; conjugate gradient methods improve convergence by choosing search directions that are conjugate with respect to the Hessian
• Quasi-Newton methods (BFGS, L-BFGS): approximate the Hessian matrix using gradient information accumulated over iterations, achieving superlinear convergence without the computational cost of computing exact second derivatives; L-BFGS is a memory-efficient variant suitable for high-dimensional problems
• Newton’s method and trust-region methods: use second derivative information (the Hessian) to construct quadratic models of the objective and take steps that minimize these models; trust-region methods add safeguards by restricting step size when the quadratic model may be inaccurate

These methods are extremely efficient for smooth, unconstrained or bound-constrained problems and are widely used in:

• Machine learning: training neural networks (backpropagation + gradient descent variants), logistic regression, support vector machines
• Statistics: maximum likelihood estimation, fitting generalized linear models
• Engineering design: optimizing designs where performance metrics are smooth functions of design parameters (e.g., aerodynamic shape optimization, structural sizing)
• Control and signal processing: tuning controller gains, filter design

Derivative-free methods are appropriate when gradients are unavailable, expensive, or unreliable (e.g., noisy function evaluations, black-box simulations, discontinuous objectives). These include:

• Nelder-Mead simplex: maintains a simplex (a geometric shape with n+1 vertices in n dimensions) and iteratively reflects, expands, or contracts the simplex to move toward better solutions; robust but slower than gradient methods
• Powell’s method: performs line searches along coordinate directions and conjugate directions without requiring derivatives
• COBYLA (Constrained Optimization BY Linear Approximations): handles nonlinear inequality constraints by building linear approximations

SciPy’s minimize provides a unified interface to these methods via the method parameter, supporting BFGS, L-BFGS-B (bounded), Newton-CG, trust-region variants, Nelder-Mead, Powell, COBYLA, and SLSQP (sequential least-squares programming for constrained problems). For scalar (single-variable) optimization, minimize_scalar offers specialized bracketing methods (Brent’s method, golden section search) and bounded methods that are more efficient than general multivariate optimizers.

When to use local optimization:

• The problem is convex (guarantees that local = global optimum)
• A good initial guess is available (from domain knowledge, previous solutions, or coarse global search)
• The objective is smooth and derivatives are available or can be approximated
• Speed and efficiency are priorities, and finding a local optimum is acceptable

The following Python functions are available in Excel:

Root Finding

Root-finding (or zero-finding) is the problem of solving equations of the form f(x) = 0, where f is a scalar or vector-valued function. While conceptually distinct from optimization, root-finding is closely related: finding a root of f is equivalent to finding a minimum of ||f(x)||², and many optimization algorithms can be adapted for root-finding (and vice versa). Root-finding is fundamental in numerical analysis and appears in countless applications: solving nonlinear equations, finding equilibrium points in dynamical systems, inverting functions, and solving implicit equations.

Scalar root-finding seeks a single value x* such that f(x*) = 0 for a univariate function f:

• Bracketing methods (bisection, Brent’s method, Ridder’s method) require an initial interval [a, b] where f(a) and f(b) have opposite signs, guaranteeing a root exists in the interval by the intermediate value theorem. These methods systematically narrow the interval and are robust (guaranteed convergence) but may be slower than open methods.
• Open methods (Newton-Raphson, secant method, Halley’s method) start from a single initial guess and use derivative information or secant approximations to iteratively improve the estimate. Newton-Raphson converges quadratically when near a simple root and the derivative is available, but may fail if the derivative is zero or if started far from a root.
• Fixed-point iteration solves f(x) = 0 by reformulating as x = g(x) for some function g and iterating xₙ₊₁ = g(xₙ). Convergence depends on g having a Lipschitz constant less than 1 near the fixed point; this method is simple but requires careful design of g.

The ROOT_SCALAR function wraps SciPy’s root_scalar, which provides access to Brent’s method (default, robust hybrid), Newton-Raphson, secant, and others via the method parameter. It automatically handles bracketing or starting guess requirements depending on the chosen method.

Multidimensional root-finding solves systems of nonlinear equations F(x) = 0 where F: ℝⁿ → ℝⁿ and x is a vector of unknowns. These arise in:

• Equilibrium problems: finding steady states of dynamical systems, chemical equilibria, market equilibria in economics
• Implicit equations: solving coupled implicit relationships (e.g., thermodynamic property calculations, circuit analysis with nonlinear components)
• Inverse kinematics: determining joint angles in robotics to achieve a desired end-effector position
• Nonlinear systems in engineering: solving coupled algebraic equations arising from discretization of PDEs, steady-state reactor design, process flowsheeting

Common algorithms include:

• Newton-Krylov methods: extend Newton’s method to systems by solving the linear system J(xₖ)Δx = -F(xₖ) at each iteration, where J is the Jacobian matrix; Krylov subspace methods (GMRES, BiCGSTAB) solve the linear system iteratively, avoiding explicit Jacobian inversion
• Quasi-Newton methods (Broyden’s method): approximate the Jacobian using rank-one updates based on observed function changes, reducing the cost of Jacobian evaluations
• Trust-region dogleg: uses a trust-region framework to ensure global convergence even from poor initial guesses

SciPy’s root provides a unified interface to these methods (hybr, lm, broyden1, krylov, anderson, and more), handling Jacobian computation (via finite differences, user-supplied analytical Jacobians, or sparse Jacobian structures) and solution diagnostics.

Reformulating optimization as root-finding: The first-order optimality conditions for an unconstrained optimization problem (minimize f(x)) are ∇f(x*) = 0, so finding a local minimum is equivalent to finding a root of the gradient. This connection is exploited in some algorithms and enables using root-finding tools for optimization and vice versa.

The following Python functions are available in Excel:

Native Excel capabilities

Excel provides several powerful built-in functions, features, and add-ins that support optimization, numerical analysis, and related computational tasks. Understanding these native capabilities helps users leverage Excel effectively and recognize when specialized tools or external libraries are needed for more demanding problems.

Core Optimization Tools

• Solver Add-in : Excel’s flagship optimization tool provides a flexible framework for linear, nonlinear, and integer programming. Users define an objective cell (the target to minimize or maximize), changing cells (decision variables), and constraint cells (restrictions on feasible solutions). Solver offers three solving engines:

  • Simplex LP: for linear programs with linear objectives and constraints; uses the simplex method and is efficient for models up to ~200 variables
  • GRG Nonlinear: implements Generalized Reduced Gradient algorithm for smooth nonlinear objectives and constraints; handles bound and general nonlinear constraints
  • Evolutionary: a genetic algorithm for non-smooth, discontinuous, or multimodal problems; slower but more robust for difficult landscapes

  Limitations: The standard Solver is designed for small to moderate-sized models (up to 200 decision variables, 100 constraints in most Excel versions; Office 365 increases limits somewhat). Large-scale problems, specialized convex optimization, or advanced features (stochastic programming, robust optimization, advanced MILP techniques) require third-party solvers. Solver’s interface requires setting up spreadsheet formulas, which can be cumbersome for complex models and lacks the expressiveness of algebraic modeling languages. Solver does provide sensitivity reports (shadow prices, reduced costs) for LP problems, enabling post-optimality analysis.

• Goal Seek : A simple root-finding tool for single-variable problems. Goal Seek adjusts one input cell to make a formula cell reach a specific target value by repeatedly evaluating the formula and adjusting the input (essentially bisection or secant method). It is ideal for quick what-if analysis: “What price do we need to charge to achieve $1M revenue?” or “What interest rate makes the NPV equal to zero?” Limitations: Only one input variable can be adjusted; no constraints; no optimization (only target seeking); uses simple iterative methods that may converge slowly or fail for poorly behaved functions.

Analysis and What-If Tools

• Analysis ToolPak : An Excel add-in providing statistical and engineering analysis tools including:

  • Regression: ordinary least-squares linear regression (fit y = Xβ + ε) with residual diagnostics, R², and confidence intervals; handles single or multiple predictors
  • ANOVA: one-way, two-way, and factorial analysis of variance for comparing group means
  • Descriptive Statistics: summary statistics (mean, median, variance, skewness, kurtosis) for datasets
  • Histogram, Sampling, Rank and Percentile: tools for exploratory data analysis
  • F-Test, t-Test, z-Test: hypothesis tests for comparing variances and means
  • Fourier Analysis: FFT for frequency-domain analysis of time series
  • Correlation and Covariance: matrices for multivariate relationships

  Use for optimization: Regression is fundamentally a least-squares problem, so the ToolPak provides parameter estimation for linear models. However, it lacks nonlinear regression, constrained estimation, robust regression, or general optimization capabilities. It’s primarily a statistical analysis toolkit rather than an optimization framework.

• Data Tables : One-way and two-way data tables systematically vary one or two input cells and record resulting formula values in a table structure. This enables sensitivity analysis, break-even analysis, and visualizing how outputs depend on inputs. For example, varying loan amount and interest rate to see monthly payment changes, or varying product price and production volume to analyze profit scenarios. Limitations: Data tables are for analysis and visualization, not optimization; they evaluate formulas at predefined grid points but do not search for optimal values; limited to two input variables; can be slow for large grids or complex formulas.

• Scenario Manager : Stores and switches between named sets of input values (“scenarios”) to compare outcomes. Useful for strategic planning: best-case, worst-case, and expected scenarios for financial forecasts; comparing alternative designs or strategies; documenting assumptions. Scenario Manager generates summary reports comparing outcomes across scenarios. Limitations: Scenarios are user-defined; the tool does not search for optimal scenarios or perform optimization; best suited for discrete alternative comparison rather than continuous optimization.

Alternative third-party add-ins

Third-party Excel add-ins significantly extend Excel’s optimization and analytical capabilities, often providing professional-grade solvers, advanced algorithms, and specialized features not available in the built-in Solver. These tools integrate directly into Excel’s interface, allowing users to leverage familiar spreadsheet workflows while accessing state-of-the-art optimization technology.

Commercial Optimization Add-ins

• Frontline Solvers  (Analytic Solver / Premium Solver Platform) — A comprehensive suite of optimization and simulation tools from Frontline Systems, the developers of Excel’s built-in Solver. Product tiers include:

  • Analytic Solver Basic/Professional/Enterprise: Extends Excel Solver with larger model sizes (up to 8,000 variables), multiple solving engines (barrier/interior-point for LP, multistart for global nonlinear optimization, branch-and-bound for MILP), and advanced features (stochastic programming, conic optimization, automatic model diagnosis)
  • Premium Solver Platform: Industrial-strength optimization with plugin engines including KNITRO (nonlinear), XPRESS, MOSEK (conic optimization), and others; supports very large-scale models (millions of variables)
  • Risk Solver Platform: Combines optimization with Monte Carlo simulation, decision trees, and stochastic programming for decision analysis under uncertainty

  These tools provide sensitivity analysis, model diagnosis (identifying infeasibility, unboundedness), automatic model transformation (converting nonsmooth models to smooth equivalents), and integration with VBA for programmatic model building and batch solving.

• OpenSolver  — An open-source Excel add-in developed at the University of Auckland, providing access to industrial-strength COIN-OR optimization engines (CBC for MILP, Bonmin for MINLP, Nomad for nonsmooth optimization) without artificial size limits. OpenSolver solves LP and MILP problems with thousands of variables and constraints—far beyond Excel Solver’s capacity—and is completely free. It maintains compatibility with Excel Solver’s model specification (constraints, variable definitions), making it a drop-in replacement. Particularly valuable for linear and mixed-integer optimization in academic, nonprofit, or resource-constrained settings. Limitations: Less polished UI than commercial tools; limited nonlinear capabilities compared to commercial solvers; no built-in simulation or stochastic programming.

• What’sBest!  (LINDO Systems) — A spreadsheet optimization add-in that allows users to specify optimization models directly in spreadsheet formulas using a natural algebraic syntax, then solves them using LINDO’s commercial optimization engines. Supports LP, QP, MILP, and nonlinear optimization with competitive performance. Includes features for sensitivity analysis, goal programming (multi-objective optimization), and model scaling diagnostics.

• Premium Solver Pro  (Frontline Systems) — Mid-tier offering between Excel Solver and Premium Solver Platform; provides significantly expanded model size limits (2,000 variables), faster algorithms, and evolutionary solver enhancements. Suitable for users needing more capacity than Excel Solver but not requiring full industrial-scale solvers.

Risk and Simulation Add-ins

• @RISK & DecisionTools Suite  (Lumivero, formerly Palisade) — Industry-leading Monte Carlo simulation and risk analysis platform for Excel. @RISK allows users to replace uncertain values in spreadsheet models with probability distributions, then runs thousands of simulation trials to quantify output uncertainty and risk metrics (VaR, CVaR, percentiles, probability of success). The DecisionTools Suite includes:

  • @RISK: Monte Carlo simulation and sensitivity analysis (tornado diagrams, spider plots)
  • PrecisionTree: Decision tree analysis for sequential decision-making under uncertainty
  • TopRank: What-if analysis identifying which input assumptions most influence outputs
  • StatTools: Statistical analysis and forecasting (regression, time series, hypothesis tests)
  • RISKOptimizer: Combines @RISK simulation with optimization to find robust solutions that perform well across uncertainty scenarios

  Widely used in finance (portfolio risk, project valuation), engineering (reliability analysis, safety margins), and operations (supply chain risk, capacity planning).

• ModelRisk  (Vose Software) — Comprehensive Monte Carlo simulation add-in competing with @RISK, offering similar simulation capabilities with additional emphasis on model auditing, distribution fitting, and advanced sampling techniques (Latin hypercube, importance sampling). Includes optimization integration and specialized tools for oil & gas, mining, and pharmaceutical applications.

• Crystal Ball  (Oracle) — Monte Carlo simulation, forecasting, and optimization suite for Excel. Provides simulation-based risk analysis, time series forecasting (ARIMA, exponential smoothing), and OptQuest optimization engine for finding optimal decisions under uncertainty. Popular in project management (schedule risk analysis) and manufacturing (capacity planning, inventory optimization).

Specialized Analysis Add-ins

• XLSTAT  — Comprehensive statistical analysis add-in providing 200+ statistical methods including regression (linear, nonlinear, logistic, Poisson), multivariate analysis (PCA, factor analysis, cluster analysis, discriminant analysis), DOE (design of experiments), quality control (SPC charts), and machine learning (classification trees, neural networks, SVM). While primarily a statistical tool, XLSTAT’s nonlinear regression and parameter estimation capabilities overlap with optimization. Particularly strong in data science, quality engineering, and academic research applications.