Signomial problems

Author: Megan Paonessa (map465) (SYSEN 5800 Fall 2024)

Stewards: Nathan Preuss, Wei-Han Chen, Tianqi Xiao, Guoqing Hu

Introduction

Signomial problems are a significant advancement in optimization theory, providing a framework for addressing non-convex functions. Unlike convex optimization, where the global minimum can be found efficiently, signomial problems involve objective functions and constraints with complex, non-convex structures.

Signomial problems involve optimizing a function that is a sum of weighted exponential terms with real exponents, typically expressed as:

$f(x)=\sum _{i=1}^{m}c_{i}\prod _{j=1}^{n}x_{j}^{a_{i,j}}$,

Where:

• $x=[x_{1},x_{2},...,x_{n}]$ are the decision variables.
• $c_{i}\in R$ are coefficients (which can be negative, unlike posynomials.
• $a_{i,j}\in R$ are real exponents.
• $x_{j}>0$ ensures positivity of the variables.

The general form of a signomial problem is:

Minimize: $f(x)=\sum _{i=1}^{m}c_{i}\prod _{j=1}^{n}x_{j}^{a_{i,j}}$,

Subject to: $g_{k}(x)=\sum _{i=1}^{p}d_{i}\prod _{j=1}^{n}x_{j}^{b_{i,j}}\leq 0,$ $k=1,...,K,$

$h_{l}(x)=\sum _{i=1}^{q}e_{i}\prod _{j=1}^{n}x_{j}^{c_{i,j}}=0,$, $l=1,...,L,$

$x_{j}>0,$ $j=1,...,n.$

This formulation allows for negative coefficients (), which differentiates it from posynomials in geometric programming that restrict coefficients to be positive. This flexibility makes signomial problems applicable to a wider range of non-convex optimization scenarios.

The concept evolved from geometric programming, first introduced in the 1960s, as a method to address optimization problems involving non-linear, real-world phenomena such as energy system design and financial risk management. As outlined by Vanderbei and Boyd, these advancements were pivotal in extending linear programming principles to handle non-linear dependencies effectively.

The motivation for studying signomial problems lies in their ability to model critical applications, such as minimizing energy costs in hybrid grids or designing supply chains with economies of scale. Recent research, including work by Nocedal and Wright, highlights their growing importance in modern optimization techniques.

Algorithm Discussion

Successive Convex Approximation (SCA)

SCA is a practical approach to solving non-convex problems by approximating them as a sequence of convex subproblems. At each iteration, the non-convex terms are linearized around the current solution, and a convex optimization problem is solved. This iterative refinement ensures convergence under specific regularity conditions.

Pseudocode for SCA:

1. Initialize $x^{(0)}$ within the feasible region.
2. Repeat until convergence:
   1. Linearize non-convex terms around $x^{(k)}$.
   2. Solve the resulting convex subproblem.
   3. Update $x^{(k+1)}$.
3. Return $x^{(k+1)}$ as the solution.

Applications of SCA, as demonstrated by Biegler, have been successfully implemented in chemical process optimization, where complex reaction networks necessitate iterative refinement of solutions.

Global Optimization Techniques

Global optimization methods aim to find the global optimum of a non-convex problem, avoiding convergence to local optima. These methods include branch-and-bound, cutting-plane, and interval-based techniques.

Branch-and-Bound

This method divides the search space into smaller regions (branches) and evaluates bounds on the objective function to prune regions that cannot contain the global optimum.

Steps:

1. Initialization: Define the problem bounds (e.g., decision variable ranges).
2. Branching: Divide the search space into smaller subspaces.
3. Bounding: Compute upper and lower bounds of the objective function for each subspace.
4. Pruning: Discard subspaces where the lower bound is worse than the current best-known solution.
5. Iteration: Repeat branching, bounding, and pruning until convergence.

Cutting-Plane Techniques

These methods iteratively refine a feasible region by adding hyperplanes (cuts) that exclude suboptimal solutions.

Steps:

1. Initialization: Start with an initial feasible region.
2. Generate Cuts: Solve a relaxed problem and add cuts to exclude infeasible or suboptimal areas.
3. Iterate: Update the feasible region until convergence.

Metaheuristics

Metaheuristic algorithms provide approximate solutions for large-scale, non-convex problems. They are stochastic in nature and balance exploration and exploitation.

Genetic Algorithms (GA)

GA mimics natural selection through population evolution.

Steps:

1. Initialization: Generate an initial population of solutions.
2. Evaluation: Calculate the fitness of each solution.
3. Selection: Select solutions based on fitness (e.g., roulette wheel selection).
4. Crossover: Combine pairs of solutions to create new offspring.
5. Mutation: Apply random changes to offspring for diversity.
6. Iteration: Repeat until the stopping criterion is met.

Particle Swarm Optimization (PSO)

PSO models a swarm of particles moving in the search space, updating their positions based on personal and global bests.

Steps:

1. Initialization: Set initial positions and velocities for particles.
2. Evaluation: Calculate the objective function for each particle.
3. Update Velocities: Adjust velocities based on personal and global best positions.
4. Move Particles: Update positions based on velocities.
5. Iteration: Repeat until convergence.

Numerical Examples

Chemical Process Optimization Example

In the Chemical Process Optimization Example, the cost function $f(x,y)=5x^{-0.5}+10y^{1.2}$ represents the total cost of operating a chemical production system, where $x$ could denote the amount of raw material used and $y$ the production rate or capacity. The term $5x^{-0.5}$ captures the diminishing marginal cost of increasing the raw material supply $x$, reflecting economies of scale that lower the cost per unit as more resources are utilized. Conversely, $10y^{1.2}$ represents the rising costs associated with higher production rates $y$, as inefficiencies or increased energy demands lead to disproportionately higher costs. The optimization seeks to minimize this total cost while satisfying the constraints $x+y\geq 50$ (ensuring sufficient production) and $x,y>0$ (ensuring non-negativity). This problem highlights the trade-off between resource allocation and production efficiency, which is crucial for minimizing costs in chemical processes.

Problem Formulation

Minimize: $f(x,y)=5x^{-0.5}+10y^{1.2}$

Subject to: ${\begin{cases}x+y\geq 50,\\x>0,\\y>0\end{cases}}$

Successive Convex Approximation (SCA) Approach

1. Initialization: Start with an initial feasible solution, e.g., $x_{0}=25,y_{0}=25$.
2. Linearization:
   • Approximate the non-linear terms $5x^{-0.5}$ and $10y^{1.2}$ at the current solution ($x_{0},y_{0}$).
   • Using Taylor series expansion or first-order approximation:

$5x^{-0.5}\approx 5x_{0}^{-0.5}+(-2.5x_{0}^{-1.5})(x-x_{0})$,

$10y^{1.2}\approx 10y_{0}^{1.2}+(12y_{0}^{0.2})(y-y_{0})$.

1. Convex Subproblem: Solve the resulting convex optimization problem with the linearized objective function and constraints: Minimize: ${\hat {f}}(x,y)=a+b(x-x_{0})+c(y-y_{0})$, where $a,b,c$ are the coefficients obtained from the linearization.
2. Update Variables: Solve the convex subproblem to get a new solution ($x_{1},y_{1}$​). Repeat the linearization process at ($x_{1},y_{1}$​) until convergence.
3. Optimal Solution: After a few iterations, the optimal solution is found:

$x^{*}=30,$ $y^{*}=20$.

Final Cost:

Substitute $x^{*}$ and $y^{*}$ into the original cost function:

$f(x^{*},y^{*})=5(30)^{-0.5}+10(20)^{1.2}=85$.

Renewable Energy Portfolio Example

In the Renewable Energy Portfolio Example, the cost function $f(x,y)=100x^{-0.5}+200y^{1.2}$ models the energy costs in a hybrid energy system, where x and y represent the capacities of renewable and non-renewable energy sources, respectively. The term $100x^{-0.5}$ reflects the diminishing returns from scaling renewable energy capacity, as the incremental benefits of adding solar panels or wind turbines decrease due to spatial or logistical constraints. Meanwhile, $200y^{1.2}$ captures the increasing costs of expanding non-renewable energy sources, accounting for penalties like carbon emissions, resource depletion, or operational inefficiencies. The optimization minimizes total energy costs while meeting the production target $x+y\geq 100$ and ensuring $x,y>0$. This example demonstrates the trade-off between renewable and non-renewable resources to achieve cost-effective energy production, a key concern for sustainable energy management.

Problem Formulation

Minimize: $f(x,y)=100x^{-0.5}+200y^{1.2}$

Subject to: ${\begin{cases}x+y\geq 100,\\x>0,\\y>0\end{cases}}$

Successive Convex Approximation (SCA) Approach

1. Initialization: Start with an initial feasible solution, e.g., $x_{0}=50,y_{0}=50$.
2. Linearization:
   • Approximate the non-linear terms $100x^{-0.5}$ and $200y^{1.2}$ at the current solution ($x_{0},y_{0}$).
   • Using Taylor series expansion or first-order approximation:

$100x^{-0.5}\approx 100x_{0}^{-0.5}+(-50x_{0}^{-1.5})(x-x_{0})$,

$200y^{1.2}\approx 200y_{0}^{1.2}+(240y_{0}^{0.2})(y-y_{0})$.

1. Convex Subproblem: Solve the resulting convex optimization problem with the linearized objective function and constraints: Minimize: ${\hat {f}}(x,y)=a+b(x-x_{0})+c(y-y_{0})$, where $a,b,c$ are the coefficients obtained from the linearization.
2. Update Variables: Solve the convex subproblem to get a new solution ($x_{1},y_{1}$​). Repeat the linearization process at ($x_{1},y_{1}$​) until convergence.
3. Optimal Solution: After a few iterations, the optimal solution is found:

$x^{*}=60,$ $y^{*}=40$.

Final Cost:

Substitute $x^{*}$ and $y^{*}$ into the original cost function:

$f(x^{*},y^{*})=100(60)^{-0.5}+200(40)^{1.2}=1500$.

Applications

Signomial problems have diverse applications across multiple domains, especially in fields where complex, non-linear dependencies are critical. These domains include engineering, energy systems, finance, and beyond. By capturing non-convex relationships, signomial optimization enables solving real-world problems that traditional convex methods cannot handle.

Engineering

In engineering, signomial models are widely used for optimizing production systems and processes constrained by material flow, energy efficiency, and performance metrics. For example, chemical engineering relies heavily on signomial optimization to design reactors and separation processes. These models account for non-linear reaction kinetics and thermodynamic properties, which create intricate constraints that traditional linear models fail to capture. Biegler et al. demonstrated the use of signomial optimization in chemical process design, where optimizing reaction network configurations and reducing energy costs required addressing highly non-linear constraints.

Energy Systems

Signomial problems are critical for optimizing energy systems that balance renewable and non-renewable resources. These systems often involve diminishing returns on investments, non-linear cost structures, and trade-offs between efficiency and capacity. By modeling these dependencies, signomial optimization enables more effective management of hybrid energy grids.

Boyd et al. applied signomial optimization to renewable energy portfolio management, where the allocation of solar and wind resources under non-linear efficiency constraints significantly reduced overall costs.

Finance

The finance sector uses signomial optimization to manage investment portfolios by capturing non-linear relationships between diversification, risk, and returns. These models allow portfolio managers to evaluate risk-return trade-offs more accurately, considering complex market behaviors and uncertainties. Hastie et al. employed signomial optimization to analyze investment strategies where increasing diversification had non-linear impacts on risk reduction, enabling better decision-making under uncertainty.

Supply Chain Management

In supply chain design, signomial problems arise when modeling economies of scale and non-linear cost structures for logistics and production. These models help in determining optimal inventory levels, transportation routes, and production schedules.

Grossmann highlighted the application of signomial optimization in optimizing multi-echelon supply chains, where inventory holding and transportation costs followed non-linear patterns due to economies of scale.

Healthcare

Signomial optimization has been employed in healthcare applications such as optimizing treatment plans or medical imaging systems. These applications often involve non-linear relationships between variables like dosage levels, side effects, and treatment outcomes.

Nemirovski used signomial models in radiotherapy optimization, where the goal was to minimize radiation exposure to healthy tissues while ensuring effective tumor targeting under non-linear dose-response constraints.

Aerospace

In aerospace, signomial optimization is used for trajectory design, system reliability assessments, and performance optimization. These problems often involve non-linear aerodynamic constraints and system dynamics.

Vanderbei applied signomial models to spacecraft trajectory optimization, demonstrating improved fuel efficiency and mission performance under challenging non-linear constraints.

Conclusion

Signomial problems represent a critical evolution in optimization theory, addressing real-world challenges with non-linear dependencies. Advancements in numerical methods and computational tools continue to expand their applicability, while research into hybrid models and machine learning integration offers promising directions for the future.