MLN1117 in Combination With Docetaxel, Paclitaxel, and Other Investigational Anticancer Agents to Treat Participants With Gastric and Gastroesophageal Adenocarcinoma

Inclusion Criteria: Part 1 and Part 2 1. Is male or female aged 18 years or older at the time of consent. 2. Has Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 within 14 days before enrollment. 3. Has adequate organ and hematologic function as evidenced by the following laboratory values within 14 days before enrollment: * Absolute neutrophil count (ANC) ≥1.5x10\^9/L. * Platelet count ≥100x10\^9/L. * Hemoglobin ≥9 g/dL (Transfusions are allowed to reach this hemoglobin level). * Serum creatinine ≤1.5 times the upper limit of the normal range (ULN) or creatinine clearance ≥50 mL/min either as estimated by the Cockcroft-Gault equation or based on urine collection (12 or 24 hours). * Total bilirubin ≤1.5×ULN. * Aspartate aminotransferase (AST) and Alanine aminotransferase (ALT) ≤2.5×ULN. 4. Female participants who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 1 highly effective method and 1 additional effective (barrier) method of contraception at the same time, from the time of signing the informed consent form through 30 days after the last dose of study drug (with the exception of those participants assigned to TAK-659, for whom the duration required is 180 days), or for as long as mandated by local labeling for docetaxel and paclitaxel, OR * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the participant, from the time of signing the informed consent form through 30 days after the last dose of study drug (with the exception of those participants assigned to TAK-659, for whom the duration required is 180 days), or for as long as mandated by local labeling for docetaxel and paclitaxel. (Periodic abstinence \[eg, calendar, ovulation, symptothermal, postovulation methods\], withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.) Male participants, even if surgically sterilized (ie, status postvasectomy), who: * Agree to practice effective barrier contraception during the entire study treatment period and through 120 days after the last dose of study drug (with the exception of those participants assigned to TAK-659, for whom the duration required is 180 days), or for as long as mandated by local labeling for docetaxel and paclitaxel, OR * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the participant during the entire study treatment period and through 120 days after the last dose of study drug (with the exception of those participants assigned to TAK-659, for whom the duration required is 180 days) or for as long as mandated by local labeling for docetaxel and paclitaxel. (Periodic abstinence \[eg, calendar, ovulation, symptothermal, postovulation methods for the female partner\] and withdrawal are not acceptable methods of contraception.) 5. Voluntary written consent must be given before performance of any study-related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the participant at any time without prejudice to future medical care. 6. Has suitable venous access for the study-required blood sampling (ie, pharmacokinetic (PK) sampling, circulating tumor deoxyribonucleic acid \[DNA\]). Part 1 only 1. Has a histologically confirmed diagnosis of advanced solid tumor, including but not limited to gastric or gastroesophageal junction adenocarcinoma. 2. Has radiographically or clinically evaluable disease. Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST), Version 1.1 is not required. 3. Is relapsed or refractory with no effective therapeutic options available. Part 2 only 1. Has a histologically confirmed diagnosis of metastatic or locally advanced adenocarcinoma of the stomach or gastroesophageal junction (Stage IIIb or IV according to International Union Against Cancer \[UICC\] tumor, node, metastases \[TNM\] classification, 7th edition). 2. Has at least 1 measurable tumor lesion per RECIST Version 1.1 by radiographic techniques (computed tomography \[CT\] or magnetic resonance imaging \[MRI\]). 3. Has receipt of 1 prior systemic chemotherapy regimen for advanced or metastatic adenocarcinoma of the stomach or gastroesophageal junction with documented progressive disease (PD). 4. Has archived or fresh tumor biopsy samples obtained during screening sufficient for Epstein-Barr virus (EBV) testing and genotyping. Exclusion Criteria: Part 1 and Part 2 1. Has received prior systemic anticancer therapies or other investigational agents within 2 weeks before the first administration of study drug or has failed to recover from the adverse drug effects of prior therapies (to ≤Grade 1 or to a level meeting inclusion criteria). For prior therapies with a half-life longer than 3 days, the interval must equal minimally 28 days before the first administration of study drug and the participant must have documented PD. 2. Has radiotherapy within 14 days before enrollment. 3. Has fasting glucose ≥130 mg/dL. Poorly controlled diabetes mellitus (glycosylated hemoglobin \[HbA1c\] \>7.0%). Participants with a history of transient glucose intolerance due to corticosteroid administration are allowed. 4. Has received strong cytochrome P-450 (CYP) 3A4 inducers/inhibitors within 7 days before the first administration of study drug or has conditions that require the concomitant use of CYP3A4 inducers/inhibitors during the course of the study. 5. For TAK-659 (Cohort A) only: Is receiving treatment with medications that are known to be inhibitors or inducers of P-glycoprotein (P-gp). Baseline lipase \>ULN. Participants not fulfilling these exclusion criteria can be enrolled in other cohorts (Part 1 only). 6. Has taken proton pump inhibitors within 7 days before the first administration of study drug or has conditions that require the concomitant use of proton pump inhibitors during the course of the study. 7. Has signs of peripheral neuropathy ≥ National Cancer Institute Common Terminology Criteria for Adverse Events (NCI CTCAE) Grade 2. 8. Has symptomatic brain metastases or brain metastases with a stable neurologic status for \<2 weeks after completion of the definitive therapy and steroids. 9. Has systemic infection requiring intravenous (IV) antibiotic therapy or other serious infection within 14 days before the first dose of study drug. 10. Has known or suspected human immunodeficiency virus (HIV) positive or hepatitis B surface antigen-positive status, or known or suspected active hepatitis C infection. Testing for these agents is not required in the absence of clinical findings or suspicion. 11. Has known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerability of orally administered study drug, including difficulty swallowing tablets; diarrhea \>Grade 1 despite supportive therapy; or prior total gastrectomy. 12. Has clinically significant comorbidities, such as uncontrolled pulmonary disease, known impaired cardiac function or clinically significant cardiac disease, active central nervous system disease, or any other condition that could compromise the participant's participation in the study. • Known impaired cardiac function or clinically significant cardiac disease includes: evidence of currently uncontrolled cardiovascular conditions (including arrhythmias, angina, pulmonary hypertension, acute ischemia or active conduction system abnormalities); current history of New York Heart Association Class III or IV heart failure; acute myocardial infarction within 6 months before starting study drug; baseline QT interval corrected for heart rate (QTc) ≥Grade 1 according to NCI CTCAE Version 4.03 criteria; or abnormalities on baseline 12-lead ECG that are considered clinically significant per the investigator. 13. Female participants who are lactating and breastfeeding or have a positive serum pregnancy test during the screening period or a positive urine pregnancy test on Day 1 before the first dose of study drug. 14. Participants with bilirubin \>ULN, or AST and/or ALT \>1.5 X ULN concomitant with alkaline phosphatase \>2.5 X ULN cannot be allocated to Cohort D (MLN1117+docetaxel) in Part 1 and are not eligible for Part 2 if they are also EBV negative. Part 2 only 1\. Has prior treatment with any of the following: * An Aurora A-targeted agent (not eligible for randomization in Cohorts B, C, or D, but eligible for Cohort A if EBV positive). * A docetaxel- or paclitaxel-containing chemotherapy regimen (not eligible for randomization in Cohorts B, C, or D, but eligible for Cohort A if EBV positive). * A spleen tyrosine kinase (SYK) inhibitor (MLN1117+TAK-659 arm only). * A phosphoinositide 3-kinase (PI3K) or serine/threonine kinase, also known as protein kinase B or PKB (AKT) inhibitor.